Apple has said its iTunes online music store sold five times more songs than rival Napster's legitimate new service in its first week of operation.
The revamped Napster service launched last month
Apple said consumers bought and downloaded 1.5 million songs during the same period that Napster reported selling 300,000 songs.
Napster re-opened for business in the US on 29 October after two years.
The original service had 60 million users but was forced to close when record companies began legal action.
The pioneering song-swapping service had been accused of copyright infringements.
A legal version renamed Napster 2.0 is now available to US customers who must pay for downloading music.
It is offering single tracks for 99 cents and albums for $9.95. There are no plans to launch in Europe.
Apple quoted figures showing it had 80% of the market for legally purchased downloads last week.
More than 17 million songs have been bought on iTunes at 99 cents each since the service was introduced in April.
The iTunes software, which has the music store integrated into it, is now available for both Macintosh computers and those who use Microsoft's Windows operating system.
Meanwhile, Penn State University in the US has reached a deal to offer thousands of students free access to Napster.
The move is aimed at stemming wide-scale online piracy on college campuses.
Students can also buy permanent downloads that can be burned to CDs or transferred to portable devices for 99 cents each.