Rage Against the Machine's appearance on BBC Radio 5 live attracted 32 complaints to the station, after the band swore four times during a live set.
Presenter Shelagh Fogarty apologised straight away, after she was heard telling a producer to cut the audio.
The BBC had asked the band and its management for an assurance that they would change the original lyrics and not use strong language on-air. It said that a specific assurance was given by the band on each occasion.
The broadcaster said that while it accepted there was a degree of risk in asking the band to perform live, reasonable steps were taken to minimise this risk.
However, Ofcom resolved the complaints given the measures taken by and assurances given to the broadcaster before the broadcast, the conduct of the band during the interview and start of the song performance, and the apologies issued.
The ever helpful RadioFail website has a recording of the audio right [link=http://radiofail.wordpress.com/2009/12/17/rage-against-the-machine-sweary-breakfast-fail/]here[/link].