The leak of the document came when the General Election campaign was at its height and its contents were denied by Ms Sturgeon, who accused the UK Government of “dirty tricks”.
Mr Carmichael, the Lib Dem MP for Orkney and Shetland, said the leak was an error of judgement.
The former Scottish Secretary made the admission as the Cabinet Office published its inquiry into the leak.
The investigation, ordered by the UK’s most senior civil servant Sir Jeremy Heywood, found that Mr Carmichael’s former special adviser Euan Roddin leaked the confidential memo to the Daily Telegraph - but he had Mr Carmichael’s permission to do so.
The leak inquiry concluded that Mr Roddin’s official mobile phone was used to call the Telegraph journalist and that Mr Carmichael “could and should have stopped the sharing of the memo”.
Neither Mr Carmichael nor Mr Roddin will take their severance payments as cabinet minister and special adviser. Generally, a lump sum, equivalent to three months of annual ministerial salary, is payable when a minister ceases to hold office.
Mr Carmichael has admitted that he should not have let Mr Roddin leak the memo and added that he accepted “full responsibility for the publication.”
He has written to Ms Sturgeon and the French Ambassador to apologise - describing the incident as a “breach of protocol.”
The Lib Dem MP added that, had he still been a government minister, he would have “considered this to be a matter that required my resignation”.
In his letter to the Scottish First Minister, Mr Carmichael said: “I wish to inform you that I am taking full responsibility for the publication of that document when I was Secretary of State.
“I accept that its publication was a serious breach of protocol and that the details of that account are not correct.”
He added: “I am clear that this was an error of judgment on my part and wish to offer you my sincere apologies for the embarrassment caused to you and the French ambassador.”
The memo was an account of a meeting Ms Sturgeon had with the French Ambassador Sylvie Bermann and the French Counsul General Peirre-Alain Coffinier in February.
The memorandum drawn up by a civil servant on March 6 gave an account of a telephone conversation with Mr Coffinier. The official asked to be “filled in” by Mr Coffinier on the meetings the Ambassador had with politicians, including Ms Sturgeon, on a visit to Scotland.
The document claimed that Ms Sturgeon had “no idea ‘what kind of mischief’ Alex Salmond could get up to and confessed that she would rather see Mr Cameron remain as PM (and didn’t see Ed Miliband as Prime Ministerial material.)”
The document then recorded the civil servant noting: “I have to admit that I’m not sure that the FM’s tongue would be quite so loose on that kind of thing in a meeting like that, so it might well be a case of something being lost in translation.”
The Cabinet Office report said the civil servant “believed that the memo was an accurate record of the conversation that took place between him and the French Consul General” but pointed out that he had already highlighted that it could have been “lost in translation”.
The Cabinet Secretary Sir Jeremy Haywood concluded that there was no reason to doubt that the civil servant recorded accurately what he thought he had heard.