A tweet by Commons Speaker's wife Sally Bercow about Lord McAlpine was libellous, the High Court in London has ruled.
Lawyers for Lord McAlpine said the tweet - which read "Why is Lord McAlpine trending? *Innocent face*"- pointed "the finger of blame" at the Tory peer during a media frenzy over allegations of child sex abuse.
Damages are to be decided at a later date, unless a settlement is reached.
Sally Bercow arriving at court
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 still stood, her lawyers said.
She said in a statement after the ruling that the tweet was intending to be "conversational and mischievous," but she hoped her case would 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 own lesson the hard way."
She added: "To say I'm 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."
The tweet was posted two days after a November Newsnight report wrongly implicated the former Conservative Party treasurer in claims about events at Bryn Estyn children's home in the 1970s and 1980s.
Lord McAlpine, who has already received six-figure payouts from the BBC and ITV, says it meant he was a paedophile who was guilty of sexually abusing boys living in care, and wants damages.
The judge said that in its natural and ordinary meaning, the tweet meant that Lord McAlpine was a paedophile who was guilty of sexually abusing boys living in care.
He added that, if he was wrong about that, he would find that it bore an innuendo meaning to the same effect.
"In my judgment the reasonable reader would understand the words "innocent face" as being insincere and ironical.
"There is no sensible reason for including those words in the Tweet if they are to be taken as meaning that the Defendant simply wants to know the answer to a factual question," he said.
Andrew Reid of RMPI Solicitors, representing McAlpine, said: "The apologies previously received from Mrs Bercow did not concede 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 unnecessary 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 accepted an offer of settlement, which Lord McAlpine made back in January.
"Mr Justice Tugendhat's judgment is one of great public interest and provides both a warning to, and guidance for, people who use social media. It highlights how established legal principles apply to social media, and how the courts take account of the particular way in which social media operates when reaching decisions on whether publications are defamatory."
During the hearing, Sir Edward Garnier QC said it would be difficult to think of a more serious meaning than the one advanced by Lord McAlpine, who was not in court.
"The tweet, by itself, suggests that 'Lord McAlpine' has done something wrong. Drawing attention to someone and then adding the expression 'innocent face' hints at wrongdoing and negates any suggestion that the tweet was a neutral query to which the defendant was looking for an answer.
"The inclusion of the words 'innocent face' was giving a nudge and a wink to readers."
He said a user of Twitter was not someone who ignored what was happening in the world - he or she, far from being a hermit or shut off from the media, was inquisitive and IT literate.
"In short, there was a prominent and salacious story in the media, and what was missing was the name of the abuser at its centre.
"Put another way, what was the tweet about, if it was not pointing the finger of blame at Lord McAlpine?"