The High Court has ruled that Sally Bercow did libel Lord McAlpine via Twitter and she has agreed to pay him damages.
Her posting appeared two days after a November 2012 Newsnight report wrongly implicated the former Conservative Party treasurer in allegations of sex abuse at Bryn Estyn children's home in the 1970s and 1980s.
Bercow has always denied that the tweet - "Why is Lord McAlpine trending? *Innocent face*" - was defamatory.
But Mr Justice Tugendhat said today: "I find that the tweet meant in its natural and ordinary defamatory meaning that the claimant was a paedophile who was guilty of sexually abusing boys living in care.
"If I were wrong about that I would find that the tweet bore an innunendo meaning to the same effect. But if it is an innuendo meaning, it is one that was understood by that small number of readers who, before reading the tweet on 4 November either remembered or had learned that the claimant had been a prominent conservative politican in the Thatcher years.
"At this stage I am not asked to find how many followers of the defendant read the tweet or understood it in the meaning I have found it bore."
Lord McAlpine's solicitor Andrew Reid said: "Lord McAlpine is pleased with Mr Justice Tugendhat's finding that Mrs Bercow's tweet was highly defamatory. Apologies previously received from Mrs Bercow did not conceed that her tweet was defamatory. Clearly she must now accept this fact.
"The failure of Mrs Bercow to admit that her tweet was defamatory caused considerable unncessary pain and suffering to Lord McAlpine and his family over the past six months. With knowledge of the judgment I am pleased to be able to say that Mrs Bercow has finally seen sense and has accept an offer of settlement which Lord McAline made back in January.
"Mr Tugendhat's judgment is one of great public interest and provides a warning to, and guidance for, people who use social media."
Lord McAlpine, who has already received six-figure payouts from the BBC and ITV, said it meant he was a paedophile who was guilty of sexually abusing boys living in care, and wants damages.
Mr Justice Tugendhat, in London, has heard that Mrs Bercow promptly tweeted her apologies, provided letters apologising for the distress caused and making clear that the underlying allegations were untrue, and made an offer to settle the case which had not been withdrawn.
Sally Bercow said in a statement: "In November 2012 I tweeted the question 'why is Lord McAlpine trending? *innocent face*'. I did not tweet this with malice, and I did not intend to libel Lord McAlpine.
"I was being conversational and mischievous, as was so often my style on Twitter.
"I very much regret my tweet and I promptly apologised publicly and privately to Lord McAlpine for the distress I caused him. I also made two offers of compensation.
"Lord McAlpine issued proceedings and the last few months have neen a nightmare. I am sure he has found it as stressful as I have. Litigation is not a pleasant experience for anyone.
"Today the High Court found that my tweet constituted a serious libel, both in its natural meaning and as an innuendo. To say I am surprised and disappointed by this is an understatement. However, I will accept the ruling as the end of the matter.
"I remain sorry for the distress I have caused Lord McAlpine and I repeat my apologies. I have accepted an earlier offer his lawyers made to settle this matter.
"Today's ruling should be seen as a warning to all social media users. Things can be held to be seriously defamatory, even when you do not intend them to be defamatory and do not make any express accusation. On this I have learned my lesson the hard way."