(FORTUNE Magazine) – The Rockefeller Room in Dallas might be the most exclusive eatery in the world. There's just one table with four place settings, and it takes decades to secure a seat for lunch. It's not the food that creates the allure: The entrees are stylish but simple, with recent offerings like butternut-squash ginger soup and pan-seared king salmon. And it's not the wine list either: There's no booze allowed. The Rockefeller Room's cachet comes from the fact that its guest list consists of the greatest collection of oil-industry power since the place's namesake, John D. Rockefeller, dined alone.

Here is where the men who control Exxon Mobil's destiny--CEO Lee Raymond and his top lieutenants--gather for lunch. Among them, the four oversee a $200-billion-a-year enterprise that is on track to be the world's most profitable company this year. Exxon owns the world's largest private collection of oil and gas reserves. It is also the largest refiner, as well as the biggest private producer of natural gas. And when the four men are all in town, they can be found in the Rockefeller Room each day from about noon to 1 P.M., going over everything from how to boost the company's already staggering production levels--it pumps more oil than Kuwait--to dealing with the perpetual PR nightmares that have made Exxon the oil company green activists love to hate.

Soon the seat at the head of the table will be available. Lee Raymond is stepping down. A legend in the energy industry for his unapologetic stance as a proud oilman, Raymond turned 65 last month, the age Exxon normally requires its executives to retire. In 2001 the board of directors extended Raymond's tenure so that he could groom a successor, a task that had been delayed by the acquisition of Mobil in 1999. That $81 billion deal was the largest industrial merger ever, and the crowning triumph of Raymond's ten-year tenure.

After 40 years at Exxon, Raymond is ready to move on. "It's time for me to do other things I'd like to do," Raymond said in a rare interview. "It's not right for me to be around here forever, not right for the other people who are here." The favorites to succeed him are part of the Rockefeller Room quartet. One is 51-year-old Rex Tillerson, a native Texan who is responsible for the upstream side of the business--finding oil and gas and getting it out of the ground. These operations generated more than 80% of Exxon's profits last year, but Tillerson doesn't have a lock on the job. Ed Galante, 52, a veteran marketer and engineer who oversees Exxon's downstream operations--the refining and chemicals businesses--is also a candidate. (The fourth man in the Rockefeller Room, executive vice president Harry Longwell, 62, is too close to retirement to be a contender.)

A decision is still months away, but the grooming has begun. The board of directors has been meeting with Tillerson and Galante since they assumed their current jobs in 2001, and both men made presentations at the board's retreat in Scotland in June. The company has also been showcasing the execs on Wall Street. In November, Galante will address Merrill Lynch's annual energy conference.

For most companies, such appearances would be everyday occurrences. Not at Exxon, where Raymond has been the lone public face. "Lee is always in control, even in a strict manner," says Oppenheimer analyst Fadel Gheit, a longtime Exxon watcher. Now, with the deliberation and caution that's a hallmark of Exxon's culture, the stage is being set for life after Lee.

Whoever wins will inherit more than just the top job at a company that ranked No. 3 on the FORTUNE 500 last year--and earned $11 billion in the first half of this year. (That's more than twice what Microsoft made and close to what Exxon earned all of last year.) As the head of the world's largest energy company, which is one of five global "supermajors" (the other four are Royal Dutch/Shell, BP, Total, and ChevronTexaco), the new CEO will have an outsized say on everything from environmental policy to human rights in the developing countries where most of the world's new oil is being found.

During his tenure, Raymond has been a conservative voice in a conservative industry. For example, instead of adopting the more conciliatory, even apologetic stance of BP's John Browne, who made his company's slogan "Beyond petroleum," Raymond enraged critics with his unabashed skepticism about the potential of alternative energy sources like wind and solar. Exxon refused to continue offering the "domestic partner" benefits that Mobil provided its gay employees. And Raymond is famously dubious about whether there's any link between fossil-fuel consumption and global warming, a stance environmental activists consider akin to membership in the Flat Earth Society.

Raymond's critics will not miss him (or vice versa). Shareholders will. Since he took over as CEO in 1993, the company's stock has beaten the S&P 500 by a wide margin. Exxon has also outperformed archrivals ChevronTexaco and Shell. What's more, a dividend that's gone up for 21 years in a row has made Exxon a favorite of individual investors, who hold half of its shares. At the annual meeting in Dallas last May, a devoted shareholder declared that Raymond deserved more than the $26 million he earned last year. "I don't envy the guy who gets the top job," says Gheit. "It's like when Michael Jordan left the Chicago Bulls."

With its leather pillows, leather couch, and a Frederick Remington-style sculpture of a horse, Rex Tillerson's office is decorated in what might be called masculine not-so-moderne. A reserved man, he is an accomplished horseman and weekend rancher. Tillerson started at Exxon in 1975 as an engineer in the oilfields of south Texas; by the late 1980s he was managing oil and gas production in much of the southwest U.S. But he really made his mark overseas, where Exxon derives more than 60% of its earnings. As head of Exxon's high-profile operations in the former Soviet Union, Tillerson was critical to the company's $12 billion effort to develop oil and gas production off Sakhalin Island in the Russian Far East. "He was extraordinary in Russia," says a former boss. "He built relationships very quickly with the Russians, which wasn't easy to do." Those relationships are now poised to pay off--by the end of 2005, the Sakhalin project should produce 250,000 barrels a day, of which Exxon owns 30%.

If Tillerson is to win the top job, he's going to have to show that Exxon can increase production 3% a year. That may not sound like much, but you've got to understand that as oil and gas are pumped out of the ground, production in a typical field drops by 5% annually. Just to stay in place, then, oil companies must raise production 5% a year. A 3% annual increase therefore requires an 8% jump. Exxon has promised such growth to the what-have-you-done-for-me-lately crowd on Wall Street. Over the past two years, however, it hasn't been able to deliver. In 2002 the company pumped 2.49 million barrels of oil a day, down from 2.54 million in 2001. Thanks to gains in operating efficiency and high energy prices, profits have stayed robust. But Wall Street is eager to see Exxon pump more while keeping production costs under tight control, since that's a sure way to make money even if oil prices fall. It's up to Tillerson to deliver.

In the first half of 2003, oil production was down slightly again. But Tillerson, along with Raymond and Longwell, insists that the $10.5 billion a year that the company spends to find new fields is about to pay off. Within the next two years, Tillerson says, "we think we will achieve production volume growth of 2% to 3% annually." In 2002 and early 2003, he notes, Exxon's output was depressed by unrest in Indonesia and Venezuela. In the OPEC countries where Exxon operates, such as Abu Dhabi and Nigeria, quotas restricted production. While the outlook for OPEC is uncertain, Venezuelan and Indonesian operations are returning to normal and crude is about to gush out of new fields in non-OPEC African countries like Chad and Angola. "Sometimes you have to wait a few years, and then these things come on and you get a surge," says Tillerson. "What the boss and the shareholders want is for us to deliver on the plans."

Like Raymond and Longwell, Tillerson has a reputation as an intense leader with a famous memory for the details of Exxon's myriad oil and gas fields. "He's pretty hard driving," says a former colleague. "If you send him an e-mail, you'll hear back within an hour. He doesn't leave it to his secretary."

Tillerson can point to some good numbers to bolster his claim to the top job. One is $9.6 billion--the amount of profit his division earned last year. Another is 22.3%, the return on capital employed (ROCE), nearly double the industry average. Moreover, first under Longwell and more recently under Tillerson, Exxon has proved adept at replacing the reserves it exhausts. In 2002, despite pumping 1.6 billion barrels worldwide, it increased its overall reserves by 1.3%, more than Shell or Chevron. Finally, the cost of finding each new barrel of oil dropped to an all-time low of 61 cents, half what it cost five years ago--a sign that Tillerson is avoiding the dry holes and other mishaps that can trip up even the best-run oil giants.

Tillerson is Texas right down to his twang; Lee Raymond is a plainspoken native of South Dakota. In this company, Ed Galante stands out as a product of the East Coast. His accent still carries traces of a boyhood in Queens and on the South Shore of Long Island, and his outgoing, affable style is a reminder of his years in marketing. Since joining Exxon in 1972, Galante has made his name working his way up the downstream side, which is not nearly as glamorous among oil people as finding the next gusher. He ran the company's huge Baton Rouge refinery from 1988 to 1992, and later managed Exxon's Central America and Caribbean operations before being tapped to serve as Raymond's executive assistant in 1995 and 1996.

A former Exxon exec calls that "the most thankless job in the world," but it gave Galante a chance to build a bond with the boss. Galante ran the executive suite when Raymond was unavailable, and had to decide when to phone the CEO in the middle of the night. As a former senior executive puts it, "Sometimes it's not much fun to wake up Lee at 4 A.M." Galante must have made the right calls, because his career took off afterward, and by 2001 he was running the entire downstream side of Exxon.

Galante also has good numbers to show off. In the first half of 2003, downstream earnings jumped fivefold compared with the same period last year. Margins widened as managers achieved the kind of gains in operating efficiency that Raymond treasures. "We have a relentless focus on constantly taking out costs," says Galante, noting that a difference in profit margins of just two or three cents per gallon can make or break the division.

Galante lacks the hard-core experience of finding black gold in the back of beyond. But he is outgoing and comfortable with the public, attributes that could prove an advantage when it comes to dealing with some of the company's softer challenges. These include lingering lawsuits from the Exxon Valdez oil spill--the worst in U.S. history--and a European boycott spurred by Raymond's stance on global warming. How many CEOs have to deal with not one but two activist groups solely focused on their company, complete with catchy T-shirts and slick websites (stopesso.com and campaignexxonmobil.org)?

To be fair, the company's public image has pretty much stunk since the days of John D. Rockefeller and the Standard Oil trust (the predecessors of Exxon and Mobil were Standard Oil of New Jersey and New York, respectively). And a BP-style marketing blitz won't get most people over their ingrained suspicion of Big Oil in general, and Exxon in particular.

If either of the candidates to succeed Raymond can address the company's tattered image, however, it's Galante. Unlike Raymond and Longwell, who bristle at criticism, the silky Galante seems unruffled by the attacks on Exxon. "At its root, we are a very large, very successful, capitalist enterprise that is truly global but American in our parentage," he says calmly. "There are people who would dislike any one of those things. But we are trying to help people understand our logic."

Maybe PR skills shouldn't play an important role in choosing the CEO. The company certainly gives a good impression of not giving a damn what others say about it. Privately, however, board members are concerned because the activist blitz shows no sign of easing up. They are keenly aware of Galante's strength as a communicator. At the same time, Tillerson's upstream credentials are compelling.

Even with Raymond still in charge, Exxon does seem to be undergoing a bit of glasnost, inviting critics like Arvind Ganesan of Human Rights Watch to speak at a corporate retreat and running ads that boast of the company's efforts to improve fuel efficiency. "We're not changing our views," Galante says, "but we are trying to do a better job of communicating them."

For now, Lee Raymond continues to preside over the Rockefeller Room. He and Longwell are closely watching the two younger men, who in turn are closely watching each other. It's a classic corporate bake-off, and the contest is too close to call. "It's obvious to everyone what's going on," says a board member of the interaction between Tillerson and Galante. "But it's subliminal, and everyone gets along." No doubt each man can't wait for lunch to be over.

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