The BBC will apologise to "licence fee payers" for the "grossly offensive" phone call made by Russell Brand and Jonathan Ross tomorrow, it announced today.
The apology will be broadcast on Radio 2 just after 10am when Ross, who has been suspended without pay for three months for the prank, would normally be hosting his weekly show.
It will be repeated just after 9pm, when Brand, who resigned as a result of the fallout from the broadcast, made on his own Saturday night show, would usually be on the airwaves.
The BBC will say that the phone call to Fawlty Towers actor Andrew Sachs's answering machine should never have been recorded or broadcast.
It will apologise to its "audiences as licence fee payers" and to Mr Sachs for the incident, in which Brand boasted he had slept with the actor's granddaughter Georgina Baillie.
The BBC is currently carrying out an inquiry into the events which led to the call being broadcast.
The prank caused the resignation of Radio 2 boss Lesley Douglas and threw the corporation into days of turmoil.
The apology will state: "On October 18, 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 week, Baillie, 23, called for Brand and Ross to be reinstated, saying the resignation and three-month suspension without pay was "out of proportion".
It was also revealed today that a second BBC Radio 2 executive has resigned in the wake of the controversy, which sparked more than 40,000 complaints.
Douglas's resignation has been followed by that of David Barber, the Radio 2 head of specialist music and compliance.
A BBC spokeswoman said: "We can confirm that David Barber has resigned from the BBC."
It is thought that Barber's job would have involved checking that content complied with BBC guidelines.
The BBC has not identified the person responsible for approving the broadcast which went out on 18 October.