The corporation issued its statement last night after abuse victim Steve Messham admitted that the man who abused him in the 1970s and 1980s was not Lord McAlpine of West Green.
The 70-year-old peer found himself at the centre of a storm of internet speculation after Mr Messham told BBC2's Newsnight he had been abused by a senior Conservative from the Thatcher era when he was a teenager at a north Wales children's home.
In a statement, the corporation said: "We broadcast Mr Messham's claim but did not identify the individual concerned. Mr Messham has tonight made a statement that makes clear he wrongly identified his abuser and has apologised. We also apologise unreservedly for having broadcast this report."
Earlier solicitors for Lord McAlpine indicated that they were preparing to sue for defamation, saying their client's reputation had been left in "tatters" as a result of the programme.
BBC director general George Entwistle moved swiftly to try to limit the damage, appointing BBC Scotland director Ken MacQuarrie to produce an urgent report into what had happened.
In the meantime there will be an immediate "pause" in all ongoing Newsnight investigations while the BBC is suspending all co-productions with the Bureau of Investigative Journalism which worked on the north Wales investigation.
Nevertheless the disclosures came as another huge blow for the corporation which is still reeling from the Jimmy Savile scandal - including a decision to drop a Newsnight investigation exposing the late DJ as a serial child abuser.
Last night's edition of Newsnight - which went out under the supervision of a senior news executive - carried a full, on air apology for the broadcast on November 2.
Today, BBC director general George Entwistle said the report should never have been broadcast but insisted he would not resign.
Mr Entwistle described the report as "unacceptable" and warned that staff involved in the programme shown last week could now face disciplinary action.
He told the BBC Radio 4 Today programme: "We should not have put out a film that was so fundamentally wrong. What happened here is completely unacceptable. I have taken clear and decisive action to start to find out what happened and put things right."
He said that he expected the report into what went wrong, being carried out by Mr MacQuarrie, to be on his desk by Sunday. "Further action will follow from that - disciplinary if necessary."
Mr Entwistle admitted the corporation was facing a "bad crisis" of trust in the wake of the Jimmy Savile scandal - including a decision to drop a Newsnight investigation exposing the late DJ as a serial child abuser.
Mr Entwistle insisted however that he had no intention of resigning, although he accepted that his future now lay in the hands of the BBC Trust.
"I am doing the right things to try and put this stuff straight. I am accountable to the Trust in that endeavour. If they do not feel I am doing the right things, then obviously I will be bound by their judgment," he said.
Tory MP Rob Wilson, who was a fierce critic of the BBC over the Savile affair, questioned whether Newsnight could survive the latest fiasco.
He said Mr MacQuarrie needed to examine whether the programme on the north Wales children's home scandal had been an "over-compensation" for what happened over Savile or a "diversionary tactic" to draw attention away from the BBC.
"I want to have the information and evidence in front of me and take a considered view of whether Newsnight should continue or not," he said.
The programme had included an interview with Mr Messham who described how used to be taken from the Bryn Estyn children's home to a hotel near Wrexham to be abused by men, including one described as a senior Conservative from the Thatcher era.
Lord McAlpine yesterday finally broke cover to issue a vehement public denial of the "wholly false and seriously defamatory" claims against him.
His solicitor, Andrew Reid, criticised the way the programme had broadcast Mr Messham's allegations without stating to whom they referred - sparking the internet storm around his client.
"They took what I think is the coward's way out. They ran the programme, trailed it, and then told everyone where to go and look for the name. In my view, that is creating the defamation," he said.
"They have done a very, very good job in severely damaging Lord McAlpine's reputation."
The disclosures are also potentially embarrassing for David Cameron who rushed to order two new inquiries into the north Wales child abuse scandal following Mr Messham's allegations, event though he was out of the country on a visit to the Middle East.
On Thursday the Prime Minister warned against a "witch-hunt" after presenter Philip Schofield tried to hand him a list of alleged paedophiles live on air on ITV1's This Morning.