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.
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 investment companies take a stake in the economic rights of players – 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.
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.”