The BBC Trust's editorial standards committee (ESC) has apologised over a joke made by Frankie Boyle which compared Palestine with a cake being "punched to pieces by a very angry Jew".
The committee, which acts as a final arbiter of appeals if complainants are unhappy with the response from BBC management, upheld a previous finding that the comment was inappropriate and offensive.
But it said that no further action is needed in the case.
Boyle made the remark on Radio 4 comedy sketch show Political Animal, broadcast in June 2008.
The Scottish comedian said: "I'm quite interested in the Middle East, I'm actually studying that Israeli army martial arts. And I know 16 ways to kick a Palestinian woman in the back.
"It's a difficult question to understand. I've got an analogy which explains the whole thing quite well: If you imagine that Palestine is a cake - well, that cake is being punched to pieces by a very angry Jew."
A complainant wrote to the BBC Executive branding the comment "disgusting" and "anti-Semitic".
Dissatisfied with the response, the complainant went to the editorial complaints unit (ECU), the next stage of the BBC's complaints process.
The ECU wrote back in December 2008 upholding the complaint, saying the use of the word Jew in the context was "inappropriate and offensive".
Arrangements were made with the production company for more effective editorial supervision of any future editions of the programme.
But the complainant told the ESC he was unhappy with the proposed actions and that the remark had gone through the editorial process "without ringing any alarm bells".
The committee said it endorsed the ECU's finding about the use of the word "Jew".
It said: "As a result, the committee wished to apologise to the complainant on behalf of the BBC for any offence the remark may have caused him and other listeners to the programme."
The committee noted Radio 4 fully informed the production company, Avalon, of the ECU's decision and made it aware that the material was unacceptable and breached the BBC's editorial guidelines.
The Radio 4 commissioning editor discussed the matter with Boyle's agent, who confirmed she had discussed the matter with the former Mock The Week star.
The committee noted that since the programme aired the BBC's Audio & Music division had reformed compliance procedures.
It also pointed out that since incidents like the Sachsgate scandal involving Jonathan Ross and Russell Brand, a significant amount of work had been carried out in Audio & Music, showing "the BBC's willingness to learn from such mistakes".
The committee was satisfied "that the breach, while serious, did not require an apology or correction from the BBC online or on air.
"In conclusion the committee stated that it was very sorry that the breach of editorial standards had occurred in this case, but was satisfied that no further action was required."
Controversial comic Boyle found himself in hot water again recently when the mother of a child with Down's syndrome criticised him for making fun of people with the condition during a gig in Reading.