Sam Allardyce says he is "deeply disappointed" to have left his role as England manager after just 67 days.
It came after a newspaper reported he had offered advice on how to "get around" player transfer rules.
Allardyce made a "wholehearted apology" and said he recognised some of his comments had "caused embarrassment".
The FA called his conduct inappropriate and said his contract was ended by mutual agreement. Gareth Southgate will be in charge for four England matches.
- Analysis: Why FA had to let Allardyce go
- Who is in the frame for England job?
- England a laughing stock - Alan Shearer
What did Allardyce do?
Allardyce, 61, was filmed by undercover reporters from the Daily Telegraph, who posed as businessmen for a meeting in which he reportedly said it was "not a problem" to bypass rules on third-party player ownership, and claimed he knew of agents who were "doing it all the time".
Third-party ownership - when someone other than the buying and selling club owns a stake in a player, typically an investor - has been described as a form of "slavery" by Michel Platini, the former president of European football's governing body Uefa. The practice is banned by the FA and world governing body Fifa.
The Telegraph investigation also claimed that a £400,000 deal was struck for Allardyce to represent the company to Far East investors and to be a keynote speaker at events - though he stressed he would have to "run that by" his employers.
Allardyce also referred to his predecessor Roy Hodgson as "Woy", making fun of his manner of speaking, criticised Gary Neville, one of Hodgson's assistants, and made comments about FA president Prince William, while describing Prince Harry as a "naughty boy".
Further details of the Telegraph's wide-ranging investigation are published in Wednesday's edition of the paper - including a claim that eight past and present Premier League managers have been accused of receiving illicit payments for transfers.
Five of the un-named eight have denied the allegations while three are yet to comment, the paper says.
'Regret' at losing dream job - the apology
Allardyce was appointed England manager in July after 22 years managing clubs including Bolton, Blackburn, Newcastle, West Ham and Sunderland.
He succeeded Hodgson following a disastrous Euro 2016 tournament to become the 14th permanent England boss, the pinnacle of a managerial career that started at Blackpool in 1994 and has taken in 467 Premier League games.
Only Sir Alex Ferguson, Arsene Wenger and Harry Redknapp have managed more games in England's top division, and prior to landing the job he had openly spoken of his desire to manage the national side.
In a statement on the Football Association website, Allardyce said it had been "a great honour" to be appointed England manager and that he was "deeply disappointed at this outcome".
"Although it was made clear during the recorded conversations that any proposed arrangements would need the FA's full approval, I recognise I made some comments which have caused embarrassment," he said.
"As part of the meeting, I was asked to clarify what I said and the context in which the conversations took place. I have co-operated fully in this regard.
"I also regret my comments with regard to other individuals."
Allardyce says he now plans to go on holiday with his family to reflect on what has happened - but intends to return to management.
He told Sky Sports he had only attended the meeting with the undercover reporters as a favour to friend and agent Scott McGarvey, who he says was hoping to land a job out of it.
'Tough decision for FA'
FA chief executive Martin Glenn said Allardyce was "distraught" to have left his role.
"He recognises that he made a terrible error of judgement," he said.
"Sam understands why the FA has had to take this tough decision."
What next for England?
After just one match in charge - a 1-0 win over Slovakia in England's opening World Cup 2018 qualifier - Allardyce becomes the national side's shortest-serving full-time manager.
The FA said it would begin its search for a new England manager while England Under-21 boss Southgate takes charge for the forthcoming World Cup qualifiers against Malta, Slovenia and Scotland plus a friendly with Spain.
Allardyce was due to name his next squad on Sunday.
What they said
"I'm angry, I'm sad, I'm staggered at the misjudgement from a guy who said this was his dream job." Ex-England striker and BBC pundit Alan Shearer.
"I've got a little bit of sympathy for him, but he gave the FA no choice." Former Wales midfielder Robbie Savage.
"If you want to be the England manager you have to be whiter than white and the Telegraph investigation shows he wasn't. This guy is being paid around £3m a year, why was he grubbing around trying to find £400,000 from somewhere?" Former FA chairman Greg Dyke.
"There's no question he brought the FA and football into disrepute and that's not acceptable. I have very little sympathy." Dyke's predecessor as FA chairman, David Bernstein.
"I don't think fans will be annoyed that he is explaining how to get around rules, I think it is more that he was chasing money around the world when the focus should have been on the England job." Former England defender Danny Mills.
BBC Sport chief football writer Phil McNulty
Sam Allardyce never made a secret of his desire and suitability to be England manager - so he will be heartbroken that the dream he harboured throughout his career is over in 67 days.
The FA will be equally dismayed at being plunged into uncertainty so soon after appointing the man it believed would signpost a bright future towards the World Cup in two years after the debacle of Euro 2016.
Sadly for Allardyce, the lack of judgement and loose-tongued approach that saw him caught in a Daily Telegraph sting meant events at Wembley on Tuesday were always heading towards an inevitable conclusion.
Some may have sympathy for Allardyce, gone after one victory in Slovakia and brought down by non-footballing matters, but the FA's statement spoke of the need for strong leadership and respect for the integrity of the game, and it clearly felt his behaviour was unbecoming of an England manager.
|England's shortest serving full-time managers (and the longest)|
|Name||Games in charge||Time in charge|
|Sam Allardyce||1 (2016)||67 days|
|Steve McClaren||18 (2006-2007)||One year, six months, 18 days|
|Kevin Keegan||18 (1999-2000)||One year, seven months, 17 days|
|Terry Venables||23 (1994-1996)||Two years, four months, 29 days|
|Glenn Hoddle||28 (1996-1999)||Two years, nine months|
|Don Revie||29 (1974-1977)||Three years, seven days|
|Walter Winterbottom||139 (1946-1962)||18 years|
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