The winner didn't actually have to download an app, either. There was a non-download entry form as well. The winner was either the downloader of the 10 billionth app, or the person who submitted an entry form immediately after the 9,999,999,999 app was downloaded.
The number of downloads in the App Store have been skyrocketing, along with device use. By the end of September 2009, the App Store was at 2 billion apps downloaded. About a year ago, in January 2010, there were 3 billion downloads. When the "countdown" began, the store was some 250M apps short, and that was just over a week ago.
In February of 2010, Apple had a different 10 billion download mark hit: the 10 billionth song song from iTunes: Johnny Cash's "Guess Things Happen That Way."
Update: The 10 billionth app downloaded was Paper Glider. It was purchased by Gail Davis of Orpington, Kent, UK. Davis will receive a $10,000 iTunes Gift Card.