Douglas submitted her resignation to BBC director general Mark Thompson in writing, acknowledging that the "last week has been a painful one for the BBC and particularly for BBC Radio 2".
She added: "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.
"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.
"It is a matter of the greatest possible sadness to me that a programme on my network has been the cause of such a controversy. I would like to take this opportunity to offer my personal apology to Andrew Sachs and his family and to the audience for what has happened."
Thompson wrote to Douglas that he accepted her resignation "with real sadness", a sentiment that "will be shared both across the BBC and the wider music industry". He said she had been "an outstanding head of programmes and then controller of Radio 2" and that her "decision to take responsibility for what has happened is an illustration of the integrity and commitment which has characterised your leadership".
In a separate statement, Thompson said: "The ultimate editorial responsibility for BBC programmes lies with producers and editorial managers. The consequences of errors of judgement are therefore more serious for managers."
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