NEW YORK (FORTUNE) - When News Corp. (Research) paid $580 million for MySpace, an Internet site for teens and young adults, some people figured that Rupert Murdoch's fascination with all things digital had once again led him to overpay for a new-media property.

Half a billion dollars was a whole lot of cash for an asset that a year earlier had been valued at $44 million, when a Silicon Valley venture capital firm bought in.

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What's more, Murdoch's track record in new media ventures was less than reassuring. He lost money buying an Internet service provider called Delphi, embraced so-called "push" technology and wildly overspent for Gemstar/TV Guide, whose on-screen TV guides were touted as the gateway to television.

This time, though, the News Corp.'s purchase of MySpace is looking like that rarest of rarities in the media world -- a much-ballyhooed acquisition where it turns out that the buyer underpaid.

MySpace has simply exploded since the deal was done last July. Measured in terms of page views, MySpace has become the second-most popular site on the Internet -- behind Yahoo! (Research), but ahead of MSN (Research), AOL (Research) and Google (Research). It has 66 million members, and about 250,000 new ones sign up each day. That's a mind-boggling growth trajectory for an Internet site that was launched less than three years ago.

"It looks like the best acquisition we've made in a long, long time," Peter Chernin, the second-in-command at News Corp., said in an interview with FORTUNE. "MySpace is the single biggest growth opportunity this company has."

You don't have to take his word for it. A social-networking site called Facebook, which attracts mostly college students, is reportedly on sale, with an asking price of $2 billion. A media-industry source says Facebook's owners turned down a $750 million bid from Viacom (Research). (Viacom declined comment.)

Facebook is a huge phenomenon -- both my college-age daughters swear by it, and it's among the top 10 sites on the Internet -- but it is not as big, by any measure, as MySpace. So if Facebook is worth even $1 billion, MySpace has to be valued at three or four times as much.

Give Murdoch and Chernin credit. One's 75, the other's 54, and yet they saw the potential of a site that, to most adults, still looks like a wild and woolly, unintelligible, almost unreadable hodge-podge of home pages, music by unknown bands, amateur videos, scantily-clad babes and unabashed self-promotion.

The best explanation of MySpace's appeal that I've read comes from not from the business world but from a Berkeley Ph.D. student and social researcher at Yahoo! named danah boyd who describes the site as the virtual equivalent of classic 1950s hangouts like the roller rink or burger joint -- a place where kids can go to escape parental (and other) authority, to try out different identities and, of course, to connect with one another. She gave a lecture about MySpace at the American Association for the Advancement of Science, of all places, that you can read here and she blogs at length about it here.

The greatest risk facing MySpace is what boyd describes as "moral panic" -- a growing backlash from parents, teachers, religious leaders and law enforcement officials, along with some media outlets. They warn the site is an inappropriate and potentially dangerous place for teens to hang out. While many MySpace users are in their 20s, people as young as 14 are permitted to sign on.

More to the point, there's no way, as a free site, for MySpace to verify the age of its users. You can read an excellent story about the backlash by clicking here.

Connecticut Attorney General Richard Blumenthal, a vocal critic of MySpace, recently wrote to News Corp. asking the company to provide parents with software to block the Web site, ban kids under 16, institute new measures against pornography and take other steps to protect children from sexual predators and inappropriate material.

When police arrested a Middletown, Conn., man who allegedly sexually assaulted an underage girl he met on MySpace, Blumenthal declared, "This case is a tragic, real-life example of why allowing children to mix with adults seeking sex is a recipe for disaster."

The critics are right when they say that sexual predators and porn stars hang out at MySpace. So do cops, politicians and clergy. That's what happens when 66 million people -- or, actually, somewhat fewer, because many people post more than one profile -- come together on the Internet or anywhere else.

In some ways, the backlash against MySpace echoes the uproar in the 1950s over comic books, or the 1960s over rock music, or the more recent crusades against rap lyrics and violent video games. What will those crazy kids do next?

But if some of the criticism is overblown, Chernin has no choice but to take it seriously. Only if MySpace is seen as an advertiser-friendly site will News Corp. be able to realize its potential. The very tricky challenge for the media giant is to somehow manage MySpace, without taming its cool factor.

"We don't want to change the cultural feel of MySpace," Chernin says.

Of the company's 280 employees, about a third work, at least in part, on online safety issues. They screen photos being submitted to the site and take down profiles of users who are found to be underage. About 220,000 profiles have been eliminated, according to a company spokesman. The company is also experimenting with software aimed at identifying pictures with a high proportion of bare skin, believe it or not.

"We need to be a leadership position about protecting minors on the Internet and, more importantly, giving the parents the tools they need to protect them," Chernin says.

In the meantime, MySpace is rushing to add more ad sales people. It had been selling its home page for about $100,000 a day. The price is moving up to about $750,000 a day, according to Chernin, who says he's been told that Yahoo! commands about $1 million for its prime real estate.

"There is more opportunity in monetizing the amount of traffic we have on MySpace than I think exists anywhere in the media business," Chernin says.

MySpace also has a lot of information about its users -- their ages, musical tastes, interests -- that could be very valuable to advertisers who want to target their products. Current advertisers include movie studios, fast-food outlets, soft-drinks and cellular phone companies. Some are getting creative. Wendy's (Research) created a MySpace page for its square burger here.

All of this -- the ramping up of ad sales, the efforts to improve online safety, the creation of new tools for users -- is happening at Internet speed.

"They've got to change the tires while they're doing 80 miles an hour," Chernin says.

Yes, it sounds like the late 1990s all over again, a time when all the media giants rushed online, often with disastrous results. It'll be fascinating to see if, this time around, News Corp. can ride the wave.

Plugged In is a daily column by writers of FORTUNE magazine. Today's columnist, Marc Gunther, can be reached at mgunther@fortunemail.com.