The wife of the Speaker has reached a settlement with Lord McAlpine after the high court found she libelled him in a tweet.
The tweet was sent while incorrect rumours were circulating online about McAlpine's connection to allegations of sexual abuse.
Bercow said she accepted ruling and agreed to an earlier settlement offer made by McAlpine's lawyers.
Her barrister told the high court that the tweet - "Why is Lord McAlpine trending? *Innocent face*" - should be regarded as a random thought with no defamatory meaning.
"There doesn't need to be a coherent meaning there. It's the sort of random thought, if one was sitting in one's room with one's family, you might just come out with but instead of speaking it, tweet it," he said.
McAlpine's barrister, Edward Garnier, who recently played a key role watering down libel reform, said the tweet referred to a mistaken link with child sexual abuse in a Welsh care home.
Britain's most senior libel judge, Mr Justice Tugendha, said 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 innuendo meaning to the same effect."
Bercow said: "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 intent them to be defamatory and do not make any express accusation. On this, I have learned my own lesson the hard way."
Bercow has been a prime target for the tabloid press, whose coverage of her outspoken political commentary and participation on reality TV shows was contrasted disapprovingly with her husband's office.
But the libel case puts the Labour supporter on the same side as the press, which has often raised concerns about the more draconian impacts of the UK's libel laws.