Two of the longest-running names in digital music, Rhapsody and Napster, are combining in a bid to grow market share and stay ahead of newer rivals like Spotify, Rdio and MOG.

Rhapsody, which is the largest U.S. digital music service with 800,000 subscribers, said Monday it would take over Napster, which is currently owned by retailer Best Buy Co Inc. Best Buy would receive a minority stake in Rhapsody on closing of the transaction, which is expected around November 30. The deal is stock based, but specific terms were not disclosed.

Even as music sales have dropped significantly over the last decade -- with fans buying fewer CDs and piracy taking hold -- many start-ups and investors have been willing to place a bet on being the music retail or distribution outlet of choice for the 21st Century. But to date, digital music sales has been dominated by Apple Inc's iTunes Music Store.

Napster's name is synonymous with the original bete noire of the music industry -- an illegal free music-sharing application founded by college dropout Shawn Fanning in the late 1990s.

Roxio Inc bought the Napster name in 2002 after the original Napster was shuttered by major music labels' lawsuits. It was later sold to Best Buy for $121 million in 2008, and is estimated to have less than 400,000 subscribers. Rhapsody described Napster as the second-largest U.S. digital music service but did not reveal how many subscribers Napster has.

This deal will further extend Rhapsody's lead over our competitors in the growing on-demand music market, said Rhapsody President Jon Irwin in a statement.

Rhapsody was spun off by Viacom Inc's MTV Networks and RealNetworks Inc early in 2010.

While Rhapsody is the current digital music service market leader, it is facing fast-growing competition from London-based Spotify which launched in the U.S. in July. The new service, which already has some 2 million paying subscribers mainly in Europe, is said to be adding thousands of U.S. users weekly, according to various industry estimates.

(Reporting by Yinka Adegoke in New York, editing by Gerald E. McCormick)