Tory MP Ben Bradley sent the message following reports that Corbyn had met with a Czechoslovakian spy masquerading as a diplomat during the Cold War.
Corbyn's legal team threatened Bradley with court action for defamation unless he issued an apology.
"I fully accept that my statement was wholly untrue," said Bradley.
"I accept that I caused distress and upset to Jeremy Corbyn by my untrue and false allegations, suggesting he had betrayed his country by collaborating with foreign spies.
"I have no hesitation in offering my unreserved and unconditional apology," he added, saying he would donate a sum to a local food bank in lieu of damages.
Labour said it was "pleased Ben Bradley has admitted what he said was entirely untrue and apologised".
The party on Friday brushed off claims by a former spy for then Czechoslovakia that veteran leftist Corbyn had knowingly cooperated with the communist intelligence agency, calling it a "ridiculous smear".
"The former Czechoslovak agent Jan Sarkocy's account of his meeting... has no credibility whatsoever," a spokesman for Corbyn said in a statement.
'Not registered'
The Sun, a right-wing tabloid opposed to Corbyn, published documents last week purportedly from the Czech State Security Archive referring to meetings between an agent and Corbyn in 1986.
The Sun said Corbyn had been given the codename "COB".
Corbyn was a relatively new Labour MP at the time, having been first elected to parliament in 1983, and was an active member of the trade union, anti-nuclear and anti-apartheid movements.
A spokesman for the Labour leader said he met a diplomat but never knowingly talked to a spy and "neither had nor offered any privileged information".
Svetlana Ptacnikova, who heads the Czech Security Forces Archive of the now defunct StB secret service was quoted in the Prague Daily Monitor saying Corbyn probably did not know whom he was meeting.
"Mr Corbyn was neither registered as a collaborator, nor does this (claim) stem from archive documents," Ptacnikova was quoted as saying.
She told CTK that if Corbyn had been an agent "his file would be in a different category".
Sarkocy, 64, who worked at Czechoslovakia's embassy in London and was expelled from Britain in 1989, told the Czech news agency CTK that he had met Corbyn repeatedly at the embassy in London.
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