A late night phone-in show on BBC local radio breached the BBC's editorial guidelines for allowing a caller to talk about his purported sex crimes.
The caller, who said he was speaking from prison, claimed he had had sex with children, that they had enjoyed it and that he had made "child sex" DVDs.
Rather than cutting him off, presenter Allan Beswick spoke to the caller for more than four minutes.
The committee described his actions as "a most serious error of judgement".
The 1 March broadcast, on BBC Radio Manchester and BBC Radio Lancashire, was considered to be in breach of guidelines "irrespective of whether the caller was actually describing his crimes or whether it was a hoax call".
"Trustees were deeply troubled by the fact that the caller had been allowed on air in the first place," said the BBC Trust's editorial standards committee.
It added that the programme's producer had only heard the last minute of the the conversation because they were busy preparing a news bulletin. Nonetheless, they did not tell the presenter to close down the call once they became aware of the call.
The edition in question was removed from the BBC iPlayer and Beswick apologised at the beginning of the following night's programme.
The committee said guidance had been issued to call handlers and that team members would be given refresher training.
A BBC spokesperson said: "This was completely unacceptable and clearly broke our strict editorial guidelines. A full, on air apology was made.
"All those involved recognise it was wrong to broadcast such offensive content.
"We have already reviewed and strengthened our procedures and provided additional training and supervision."