Amber Rudd has quit the cabinet and surrendered the Conservative whip, saying not enough effort is going into getting a Brexit deal.
The work and pensions secretary told the BBC there were no “formal negotiations” taking place with the EU, just “conversations”.
Ms Rudd said up to 90 per cent of government time was spent preparing for an “inferior” no-deal option.
Chancellor Sajid Javid said he was “saddened” by the resignation.
He told the BBC’s Andrew Marr Show he did not agree that the government wasn’t putting serious effort into getting a new deal with the EU.
Downing Street has announced that Environment Minister Therese Coffey will replace Ms Rudd as work and pensions secretary.
A No 10 spokesperson added that “all ministers who joined the Cabinet signed up to leaving the EU on 31 October, come what may”.
Ms Rudd’s resignation comes as PM Boris Johnson is reportedly considering defying a new law aimed at forcing him to seek an extension to the Brexit deadline if he fails to secure parliamentary approval on an exit agreement by 19 October.
A senior No 10 source told The Sunday Times: “If there isn’t a deal by the 18 [October] we will sabotage the extension.”
Speaking on the Andrew Marr programme, Ms Rudd urged Mr Johnson to obey the law.
According to the Sunday Telegraph, Downing Street says Mr Johnson “does not share the rebel interpretation” of the new law – expected to gain royal assent next week – which was drawn up by cross-party MPs, including 21 Tory rebels who have since been ousted from the party.
It is thought that Mr Johnson believes he could legally disregard some or all of the bill’s requirements – a move that could see him hauled before the courts and potentially facing prison.
This, in turn, could trigger an emergency judicial review by the Supreme Court next month, leading to a court showdown over whether the UK would be able to leave the EU with no deal on 31 October.
‘This government will carry on’
Mr Johnson wrote in the Mail on Sunday and Sunday Express that on Monday he will offer Labour leader Jeremy Corbyn “one last chance” to agree to an early election.
If Mr Corbyn refuses, Mr Johnson said “this government will simply carry on”.
He added that he would work “tirelessly” for a deal, but the government would still prepare to leave the EU on 31 October “whatever happens”.
Labour said Ms Rudd’s resignation showed the government was “falling apart”.
The MP for Hastings and Rye, who supported Remain in the 2016 referendum, said her resignation had been “a difficult decision”.
“I will be considering my position – whether I will stand as an independent Conservative should there be an election coming up,” she told the Sunday Times.
In her resignation letter to PM Boris Johnson she said: “I joined your cabinet in good faith: accepting that ‘No Deal’ had to be on the table, because it was the means by which we would have the best chance of achieving a new deal to leave on 31 October.
“However I no longer believe leaving with a deal is the government’s main objective.”
Her resignation comes after a week of setbacks for the prime minister, when a cross-party group of MPs seized control of the Parliamentary agenda.