The Beatles' Apple Corporation was seeking financial damages and court orders to stop Apple Computer using the "apple" marks in connection with the website that allows iPod users to download and save thousands of songs.
Apple Corporation - owned by former Beatles Paul McCartney, Ringo Starr and the widows of bandmates John Lennon and George Harrison - had accused the US company of breaching a trademark agreement between the firms by promoting music products on the Internet.
The verdict could cost Apple Corporation more than three million pounds ($A7.25 million) once all bills have been finalised.
On Monday, it agreed to pay Apple Computer's costs, so far estimated at about two million pounds (A$4.8 million).
Apple Computer is the market leader for music downloads, with around three million songs being downloaded each day from its iTunes service.
Judge Edward Mann, a self-confessed iPod owner, granted The Beatles' firm permission to appeal against his ruling.
"With great respect to the trial judge, we consider he has reached the wrong conclusion," said Apple Corporation manager Neil Aspinall.
"We felt that during the course of the trial we clearly demonstrated just how extensively Apple Computer had broken the agreement. We will accordingly be filing an appeal."
Apple Computer's chief executive officer Steve Jobs said: "We are glad to put this disagreement behind us.
"We have always loved The Beatles, and hopefully we can now work together to get them on the iTunes Music Store."
1991 peace deal
The long-running spat dates back to the 1980s and the two companies reached a peace deal in 1991, under which Apple Computer paid the Apple Corporation US$26 million dollars (A$34 million), which forbade each side entering the other's exclusive "field of use" of the Apple name.
London-based Apple Corp claimed the computer company was in breach of the agreement, which gave Apple Corps the exclusive right to use "apple" marks for the record business, the firm's lawyer Geoffrey Vos told the court in March.
The US company had exclusive rights to do so for the computer, telecommunications and data processing industry.
The peace had held between the two firms until the iPod was launched in 2001.
The 1991 agreement was arguably drawn up to cover compact discs and tapes, well ahead of technological developments such as iTunes.
Mr Mann said in his ruling that the deal moved the boundaries between the exclusive fields so that Computer was entitled to a wider range.
But he said corporation retained "musical content" as its exclusive preserve.
"I conclude that the use of the Apple logo in the notional LCD display (of the iTunes Music Store) does not suggest a relevant connection with the creative work," he said.
The iTunes Music Store was "a form of electronic shop" and proved to be "enormously successful", he added.
Mr Vos had told the court that calling the iPod download system a mere electronic device was a "perversion" of the constraints in the 1991 deal. He argued that the US firm had violated the deal by selling music online.
Mr Vos showed the judge how to download from the iTunes website. He chose the 1978 hit "Le Freak" by Chic, which filled the courtroom with disco beats as he showed how many times the "apple" logo appeared on the screen as he carried out the download.
Apple Corp is represented by a green Granny Smith apple, while the computer firm's original logo was a multi-coloured, striped apple with a chunk missing. Its new logo is grey.