The incorrect claims against the Tory peer came about because of the botched Newsnight investigation into allegations surrounding a care home in Wales.
Lord McAlpine said in a statement: "I am delighted to have reached a quick and early settlement with the BBC.
"I have been conscious any settlement will be paid by the licence-fee payers, and have taken that into account in reaching agreement with the BBC.
"We will now be continuing to seek settlements from other organisations that have published defamatory remarks and individuals who have used Twitter to defame me."
A BBC spokesman said: "The BBC has agreed terms with Lord McAlpine to settle his claim of libel against the corporation.
"The settlement is comprehensive and reflects the gravity of the allegations that were wrongly made."
The former Conservative Party treasurer's solicitor has said will be taking legal action against a large number of people who linked his name with unfounded allegations of child sex abuse.
Lawyer Andrew Reid said legal letters would be sent not only to broadcasters – including ITV's This Morning programme – but also to individuals who mentioned the peer's name on the internet, particularly Twitter.
Well-known people including Speaker's wife Sally Bercow and journalist George Monbiot were on a "very long list" of tweeters who mentioned Lord McAlpine on the micro-blogging website, said Mr Reid, warning their messages could end up costing them "a lot of money".
Mr Reid called on anyone who thought they may have defamed the peer to contact him to reach a settlement, warning this may be the cheapest outcome for them.
He said Lord McAlpine had been contacted by many members of the public urging him to take action to end the phenomenon of "trial by Twitter", which has seen users of the website send out defamatory comments which can quickly be distributed to millions of people, with apparent impunity.
Asked on BBC Radio 4 whether Lord McAlpine was considering action after that against the BBC, Mr Reid said: "We are beyond the considering at this point. Very sadly, we are going to have to take action against a lot of people.
"The next person on our list is in fact the This Morning programme, run by ITV, where Phillip Schofield managed to embarrass the Prime Minister ... and then destroy my client's reputation. What he did really was very, very low, and I'm amazed it was allowed, absolutely amazed.
"It sent everyone on to the internet."
Last night ITV's director of television admitted Schofield was wrong in confronting David Cameron with a list of alleged paedophiles on live TV, saying the action was "misguided".
In an interview on ITV News, Peter Fincham said he would be responding to letters from Lord McAlpine and Tory MP John Whittingdale "very quickly".
TV regulator Ofcom has received 415 complaints about last week's incident and has launched an investigation.
Mr Reid said the "long list" of those he thinks have defamed Lord McAlpine would get a "letter before action", and would have 48 hours to respond.
Twitter "is not a place where you can gossip and say the nastiest things possible with impunity", said Mr Reid.
He said Lord McAlpine has already had two apologies from Mr Monbiot. The solicitor said: "There are a lot of other people who have done exactly what he has done and they really need to come and sort this out."
He said people need to realise that Twitter "is not just a sort of closed gossip coffee-shop among a few friends".
Mr Reid added: "It goes out to hundreds of thousands of people and you must take responsibility. The public are fed up with it. We are being pushed and pushed to get on and actually end trial by Twitter."