Ross and Brand have both apologised for their behaviour
BBC Radio 2 has broadcast an apology about the lewd phone calls made by Russell Brand and Jonathan Ross to Fawlty Towers actor Andrew Sachs.
The apology was aired at 1003 GMT when Ross's show would have gone out, and a fuller version was broadcast again at 2113 GMT during Brand's former slot.
The BBC Trust said the "serious breach of editorial standards" necessitated an on-air apology to licence fee payers.
A second senior Radio 2 executive has now resigned in the wake of the row.
The first on-air apology, approved by the Trust's Editorial Standards Committee, said: "On 18 October, the BBC broadcast an exchange between Russell Brand and Jonathan Ross on the Russell Brand show on Radio 2.
"This concerned the actor Andrew Sachs and his granddaughter, Georgina Baillie.
"Some of this exchange was left on the voicemail of Mr Sachs.
"The conversation was grossly offensive and an unacceptable intrusion into the private lives of both Mr Sachs and Ms Baillie.
"It was a serious breach of editorial standards, and should never have been recorded or broadcast.
"The BBC would like to apologise unreservedly to Mr Sachs, Ms Baillie and to our audiences as licence fee payers."
This was later amended after Mr Sachs contacted the BBC to say the apology should also mention his wife and family.
For the second broadcast, the last sentence was changed to: "The BBC would like to apologise unreservedly to Mr and Mrs Sachs, Ms Baillie and their family, and to our audiences as licence fee payers."
A BBC spokesman said: "Mr Sachs asked us to specifically include mention of his wife and family in the second broadcast apology at 9.03 pm [the time at which the Russell Brand Show was broadcast] and we were happy to do this."
Music boss resignation
Sachs, 78, was upset after Brand and Ross left a series of lewd messages on his voicemail as part of a pre-recorded show taped on 16 October.
The pair made obscene comments about the actor's 23-year-old granddaughter Georgina Baillie during four separate phone calls.
Ross swore and said Brand had slept with Sachs' granddaughter.
The BBC confirmed David Barber's resignation on Friday
After thousands of people complained about the broadcast following media reports, Brand said in a statement he took "complete responsibility" for the incident.
Ross released his own statement, saying he was "deeply sorry" and regretted the "upset" he had caused.
Radio 2 executive David Barber, the station's head of specialist music and compliance, who would have been involved in checking the content of programmes prior to transmission, has resigned over the incident.
His departure follows that of Radio 2 controller Lesley Douglas last month.
Brand resigned from Radio 2 in the wake of the row, while Ross was suspended without pay for three months.
It is believed his suspension from his radio and TV talk shows and from Film 2008 will cost the entertainer more than £1 million.
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