Radio 2 Controller Lesley Douglas has resigned over the Andrew Sachs row, the BBC said today.
The row has already claimed the scalp of Russell Brand, and Jonathan Ross has been suspended.
Douglas offered her "personal apology to Andrew Sachs and his family and to the audience for what has happened".
She said: "Over the 23 years of my career at the BBC I have enjoyed a deep love and respect for both the audience and the BBC.
"The events of the last two weeks happened on my watch. I believe it is right that I take responsibility for what has happened."
Meanwhile the BBC Trust said it was "dismayed" by what had occurred in the Radio 2 broadcast.
There was no reference to the future of Ross in a statement released by the Trust's chairman, Sir Michael Lyons.
The statement said: "The BBC Trust represents licence fee payers and on their behalf has a responsibility to safeguard high standards of BBC broadcasts.
"The Trust is dismayed both that the offensive comments broadcast on the Russell Brand Show on 18th October fell so far short of audiences' legitimate expectations and by the deplorable intrusion into the privacy of Mr Sachs and his granddaughter."
It added: "On behalf of the BBC, the Trust offers a full and unreserved apology to Andrew Sachs, Georgina Baillie and the rest of his family. The Trust extends its apologies to licence fee payers as a whole."
Complaints about the incident have reached 35,700 and the prank has been condemned by the Prime Minister downwards.
Brand, 33, and Ross, 47, were suspended by the BBC yesterday, 11 days after the offensive phone call was broadcast on Radio 2.
Douglas, Controller of BBC Radio 2, 6 Music and Popular Music, was reported to be prepared to resign if members of her production staff were sacked.
The Controller took over Radio 2 in 2004 and has lured thirty-something listeners to a station once known for its more mature audience.
She said the last week had been a "painful one for the BBC and particularly for BBC Radio 2".
She said: "It is with enormous regret that I have decided to resign as Controller of BBC Radio 2, BBC 6 Music and of Popular Music. This is my decision alone."
BBC Director-General Mark Thompson said he accepted the decision with "real sadness".
Mr Thompson reported to a special meeting of the BBC Trust's editorial standards committee this afternoon.
He presented the findings of BBC management's investigation into how the comments were cleared for broadcast.