(FORTUNE Magazine) – No company could be more Spanish than Telefonica, the country's national phone giant. Walk down a street in Madrid, and sooner or later you'll stumble across Telefonica engineers wiring the guts of the city. Surf the Web in Barcelona, and before long you're bound to click on Ole, the locally based portal that Telefonica recently bought. From fixed-line phones to cyberspace, Telefonica connects Iberia--a point made graphically by a vintage mural at the group's head office on Madrid's swanky Gran Via. As you admire the trade winds painted across the atrium wall, it's easy to assume that Telefonica is just another quaint old Latin monopoly.

Think again. Since February 1997, when the government sold its remaining 21% stake, Telefonica has undergone the most radical shakeup of any former state telco in Europe. Its chairman and CEO, Juan Villalonga--a 46-year-old boardroom bruiser recruited from Bankers Trust--has slashed the payroll, revamped the corporate structure, and forced Telefonica's famously conservative middle managers to heed the American gospel of shareholder value. Now Villalonga aims to catapult Telefonica--currently 193 in the FORTUNE Global 500--onto the world stage as one of the planet's top five telecom companies.

Well, sure--isn't that the kind of market-pleasing guff that all heads of European telcos spout these days? It is, but with a key difference. Unlike his rivals north of the Pyrenees, Villalonga can already point to some impressive-sounding international numbers. Last year 26% of Telefonica's $18.2 billion in revenues came from abroad--a far higher proportion than any other major European carrier can claim. More than 50% of Telefonica's 37 million fixed lines are located outside Spain, as are 54% of its 14.4 million cellular phone subscribers and 86% of its 2.3 million pay-TV customers. Compared with its competitors, Telefonica is "very far ahead" on the international trail, says Jane Fraser, a consultant at McKinsey & Co. in New York City.

Indeed, in a recent McKinsey study, Fraser identifies Telefonica as a model for building a global firm. That's stretching a point, for in most world markets Telefonica has yet to leave much more than its calling card. In Europe, apart from its Spanish and Portuguese assets, Telefonica controls the third-ranked operator in Austria, plus a telecom joint venture in Rome, and that's about it. Worse, Telefonica is still smarting from the ill-advised signing in June of Martin Bangemann, the outgoing EU telecom commissioner, who was supposed to open doors across the Continent. The rest of the region cried foul at Bangemann's indecently swift defection, and Telefonica's board was forced to freeze the appointment (it's still on ice). Meanwhile, in North America, Telefonica is barely out of the starting gate, despite plans to launch an Internet service aimed at Hispanics later this year. As for Asia, call it terra incognita; Telefonica hasn't even begun to chart a course there.

Which leaves Latin America, the reason McKinsey is so excited about Telefonica. The company's rollout there is unmatched by any Spanish venture since the 16th century, when the conquistadors sailed forth "to serve God and the King, and get rich." Telefonica is big in Brazil, where its telecom treasures include the main fixed-line operator in Sao Paulo, the leading mobile phone business in Rio de Janeiro, and the principal carrier in the state of Rio Grande do Sul. In Argentina it holds 51% of the southern region's monopoly operator--a franchise that covers the lucrative financial district of Buenos Aires. In Chile, it's the leading shareholder in the former national telco. In fact, in virtually every country between Tijuana and Tierra del Fuego, Telefonica is the No. 1 or No. 2 player in the local market.

All told, notes Antonio Viana-Baptista, CEO of Telefonica's international arm, the group has invested $10.9 billion in Latin America (including almost $6 billion in Brazil), for a continent-wide market share of around 40%. Says Villalonga: "There is only one global operator in the telecom market in Latin America, and that is Telefonica. The rest are far behind."

Yet despite such imperial flourishes, what's really smart about Telefonica's Latin American expansion is how the company has generally held its conquistador reflexes in check. As McKinsey's Fraser puts it: "They've taken a prudent and wise risk-management approach [and] have cautiously learnt as they've gone along." On the ground, that has meant resisting the temptation to expend huge cash resources to blaze a solo trail to Patagonia--a foolhardy strategy on a continent with (until recently) heavily interventionist regimes, not to mention some painful sensitivities about all-conquering Spanish invaders.

Instead, Telefonica has been adept at selecting multiple local partners with an inside track to government, thus ensuring that the company was in a strong position when Latin American telecom markets began to deregulate seriously in the mid-1990s. In another astute move, Telefonica has negotiated agreements with other South American communications companies to sell them its managerial expertise and advanced technology for a fee. Between January and June of this year, for example, Telefonica Internacional took in $73 million in management fees, a 7.8% increase on the same period last year.

Now Telefonica is ready to sew together its painstakingly assembled empire. Over the next two years Telefonica and its U.S. minority partners (submarine-systems company Tyco International and telecom operator IDT Corp.) will invest $1.7 billion to lay an undersea fiber-optic cable that will connect Florida to almost all of Latin America.

So the challenge for Telefonica becomes how to leverage this unrivaled regional network. The obvious solution is to use Latin America as a strategic bargaining chip to build global alliances; as Christopher Neal, Latin American expert at Pyramid, a Cambridge, Mass., telecom consultancy, points out, "anyone who's really serious about entering Latin America had better have either a pretty good strategy for competing against Telefonica or a way to court them."

But that raises another question: Why would any global telecom giant be serious about Latin America right now? Consider Telefonica's own less-than-stellar current performance in the region. In the six months to June, Telefonica Internacional registered operating profits of just $57.2 million, barely higher than those of the same period last year. At the same time, Telefonica's operating expenses in Latin America rose 41%, to $221 million. It gets worse: In September, Telefonica announced that it had set aside $321 million in the second quarter as a provision against regional losses.

What's draining the Latin American balance sheet is an economic downturn that has been exceptionally severe in the key markets of Argentina and Brazil. Telefonica's income from its Argentinian investments, for instance, fell 24% between January and June compared with the same period last year.

The company contends that these financial setbacks are temporary and a price well worth paying to gain a foothold in the Latin American market. And so far the stock markets agree. Since privatization, Telefonica's stock has risen around 130% and, despite losing some ground in August, shows no sign of imminent collapse. While much of the credit is due to Villalonga's ruthless corporate downsizing--he has cut 10,000 jobs from the payroll in the past year alone--no company with Telefonica's increasingly heavy international exposure could keep pushing the stock north without a compelling foreign story to sell. And the story coming out of Latin America, notwithstanding the recession, is one of explosive market growth.

Take mobile phones. Despite falling consumer demand in the rest of the economy, Viana-Baptista still expects to double the number of Telefonica-managed cellular subscribers in Latin America to around seven million by the end of 1999. In fixed lines, too, Telefonica is racing to keep up with the market. Case in point: Sao Paulo. Telefonica has increased the number of phone lines there by a third since winning the franchise last year. "There is so much pent-up demand," enthuses the permanently jet-lagged Viana-Baptista, who reckons to spend 80 hours a month on planes, keeping tabs on his sprawling estate.

You hear the same line at Interactiva, Telefonica's Internet arm. Juan Perea, Interactiva's 36-year-old CEO, says that around 80% of Interactiva's ISP revenues now come from Latin America, as do almost two-thirds of its portal turnover, thanks to a regional spree that has seen the unit buy leading Internet businesses from Mexico to Argentina. Interactiva's sales pitch? "Local differences in Latin America are huge," says Perea. "You can't give Argentinian content to a Mexican, because he doesn't give a damn about what's going on in Buenos Aires."

Nobody's expecting Interactiva to break even soon. Most of its $10 million in regional revenues have come from low-margin ISP subscriptions rather than from advertising and e-commerce, where the potential profits lie.

But when you look at the growth projections, it's a different story. International Data Corp. estimates that total e-commerce spending across Latin America will surpass $8 billion by the end of 2003, compared with less than $167 million in 1998, as the number of regional Internet users more than doubles to 19.1 million.

That prospect is creating excitement about Interactiva's IPO this fall, when the unit will float around 30% of its stock in New York and Madrid. "This one is going to be big," predicts Manuel Torres, an analyst at Madrid brokerage Ibersecurities. How big exactly? Market rumors suggest the share issue could be valued as high as $3 billion.

Telefonica, of course, isn't the only company with its sights on Latin America. Just about every Internet and telecom heavyweight you care to name, from AT&T to Yahoo, is casting hungry eyes south of the Rio Grande. What's puzzling is that thus far, Telefonica has failed to take its pick from this crowd and identify the right global strategic partner. The mystery deepens when you recall that last year Telefonica signed a deal with MCI WorldCom to pursue joint venture opportunities around the world. But since then the silence has been deafening. Raise the subject of MCI WorldCom with Telefonica executives in Madrid, and they tell you to call Mississippi. But when FORTUNE tried to obtain MCI WorldCom's side of the affair, we were fobbed off with a bland restatement of the partnership's mission goals. Says Felipe Gomez Serrano, an analyst at Madrid securities house Safei: "The alliance with MCI WorldCom doesn't have any future unless both sides make some serious agreements." They probably won't, because what MCI WorldCom has to offer Telefonica--access to North America's 30 million-strong Hispanic population--is much less valuable than it seems. First, this market is already saturated with competitors; second, it's not that big, compared with Telefonica's Latin American portfolio; and in any case, does it really help Telefonica realize its global ambitions?

No one would disagree with Villalonga when he says Telefonica is now "the undisputed [telecom] leader in the Spanish- and Portuguese-speaking world." What Telefonica hasn't yet found is a way to break out of its (admittedly huge) linguistic and regional box. But given the distance these latter-day conquistadors have traveled in the past two years, don't bet against them.