The purchase price of $2.65 per share in cash represents an almost 100% premium on Napster's closing price of $1.36 on Friday.
Best Buy, which is one of the largest retailers of CDs, also offers digital subscription services similar to Napster.
The two companies are hoping, with their combined capabilities, they will have a better chance of competing with Apple's dominant iTunes service and its music players.
Brian Dunn, president and chief operating officer of Best Buy, said: "This transaction offers Best Buy a recognised platform for enhancing our capabilities in the digital media space and building new, recurring relationships with customers.
"Over time we hope to strengthen our offerings to consumers, who we believe will increasingly seek devices and solutions that enable them to access their content wherever, whenever and however they want."
Chris Gorog, Napster chairman and chief executive, and key members of senior management of Napster will continue to head Napster post acquisition.
Gorog said: "We believe Best Buy will be an ideal partner for Napster and are very excited by the benefits that this transaction delivers to our shareholders, partners and employees.
"We are looking forward to combining our digital media capabilities with Best Buy's resources and global network to extend our digital content platforms."
Napster, which recently launched one of the world's largest MP3 stores, had fiscal 2008 revenue of $127.5 m, with a loss of $16.5m. This was, however, a 15% increase over the prior fiscal year when losses hit $36.8m.
The digital music company has approximately 140 employees, with its headquarters in Los Angeles.
Best Buy, which in May bought a 50% stake in UK mobile phone retailer Carphone Warehouse for £1.1bn, said it does not plan to relocate Napster's headquarters or to make significant changes in personnel.
The transaction, which is subject to customary closing conditions, is expected to close before the end of the calendar year.
Napster is the best-known name in the world of music downloading, but was once the scourge of record companies all over the world because it allowed anyone with internet access to swap their record collections with other music fans for free.
The company was a huge success in the late 1990s racking up 70m users, but after legal action from record companies the service was suspended in 2001 and eventually closed after bankruptcy proceedings in 2002. It was bought in 2003 by software company Roxio.