Here we go. The race for global dominance in music downloading is on, although Apple (NASDAQ:AAPL) comes as close as you can to claiming it's already over. One week after launching in Europe, Apple said it has sold more than 800,000 songs across the United Kingdom, Germany, and France.

CEO Steve Jobs was less than humble about Europe's appetite for Apple, proclaiming in a statement that iTunes is Europe's top online music store and noting that it had trounced competitor On Demand Distribution (OD2) in last week's downloads. Jobs has a right to boast, of course. iTunes and Apple's iPod have been huge hits in the U.S. And the early returns for iTunes in Europe could foreshadow further dominance in digital music for the firm.

But it's way too early to proclaim Apple the winner. As fellow Fool Seth Jayson has pointed out, Apple's competition in Europe is fierce. Besides OD2, which was recently acquired by Loudeye (NASDAQ:LOUD), there's Roxio's (NASDAQ:ROXI) Napster, which launched in Europe last month. Loudeye is potentially the more dangerous to Apple, for it has established a partnership with Microsoft (NASDAQ:MSFT) to help others build competitive digital music stores, kind of like what OD2 did for Coca-Cola (NYSE:KO).

There are other hurdles, too. According to a report, Apple has yet to secure licenses to distribute music from roughly 800 independent labels representing many of Europe's most popular bands, including Jaxx, Franz Ferdinand, and the White Stripes. (I highly recommend the White Stripes' single "Seven Nation Army," by the way.) Till that treasure trove is made available to at least one of the major services, the digital music market in Europe is up for grabs.

That's key for Apple, because while digital downloading may not be much of a profitable business, iTunes is the engine that drives iPod sales. Bringing the best possible range of downloadable songs to consumers through iTunes Europe could prove crucial to keeping consumers on the Continent buying iPods.

Sorry, Steve, this battle is just beginning.

Are you among the digital denizens with thousands of tunes stored on your iPod? Share your favorites with other Fools at the iTunes Music Store Playlists discussion board. Think iTunes will wipe out the competition in Europe? Tell us at the Apple discussion board.

Fool contributor Tim Beyers had his iTunes collection blaring while writing this take in his downstairs office. He didn't want to disturb his kids. Tim owns no shares in any companies mentioned, and you can view his Fool profile here.