After the ruling by Mr Justice Tugendhat in Lord McAlpine's favour, Mrs Bercow said: "I have accepted an earlier offer his lawyers made to settle the matter."
The amount of damages was not disclosed.
Her posting appeared two days after a Newsnight report last November wrongly implicated the former Conservative Party treasurer in allegations of sex abuse at Bryn Estyn children's home in the 1970s and 1980s.
Mrs Bercow denied that the tweet - "Why is Lord McAlpine trending? *Innocent face*" - was defamatory, but Lord McAlpine, who has already received six-figure payouts from the BBC and ITV, said it pointed "the finger of blame" during a media frenzy.
Today the judge agreed and said it meant Lord McAlpine was a paedophile who was guilty of sexually abusing boys living in care.
Mrs Bercow said later: "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 been 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'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.
"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 own lesson the hard way."
Speaking for Lord McAlpine, Andrew Reid of RMPI Solicitors said he was pleased with the judge's finding that the tweet was highly defamatory.
"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."
The judge said that Mrs Bercow's followers on Twitter - who numbered 56,000 - were probably largely made up of people who shared her interest in politics and current affairs and knew, by the time of her post, the elements of the story told on Newsnight.
It was not necessary for them to have any prior knowledge of Lord McAlpine as a leading politician of the Thatcher years to link the tweet naming him with the Newsnight allegations as it identified him by his title.
"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."
The reader would reasonably infer that Mrs Bercow had provided "the last piece in the jigsaw".
At an earlier hearing, Lord McAlpine's counsel Sir Edward Garnier QC said Mrs Bercow was not "some kitchen table blogger" and one would have to have been "a moron in a hurry" or an "anchorite in a sealed cave" not to have known the circumstances surrounding the tweet.
Mrs Bercow's QC, William McCormick, said the question posed by the tweet was entirely neutral, while the "innocent face" was merely an indication that it should be read with a deadpan emphasis or tone, comparable with stage directions such as sotto voce or a notation on a musical score.