On Monday, Justice Edward Mann ruled that the Apple Computer used the Apple logo in association with its store, not the music, and therefore was not in breach the existing contract.
The ruling means iPods and iTunes will still be able to carry the Apple name and logo.
Apple Corps, which sought damages and an injunction baring its rival using the Apple logo in its music operations, will appeal, according to the report.
Justice Mann ruled iTunes was "a form of electronic shop" and not involved in creating music.
"I conclude that the use of the apple logo ... does not suggest a relevant connection with the creative work," he wrote in his judgment.
"I think that the use of the apple logo is a fair and reasonable use of the mark in connection with the service, which does not go further and unfairly or unreasonably suggest an additional association with the creative works themselves."
Apple Corps now must pay Apple Computer's legal bill, estimated at £2m. The judge reportedly refused an interim payment of £1.5m pending further hearings.