Jo Johnson has quit as transport minister and called for the public to have a fresh say on Brexit.
The MP, who is Boris Johnson's brother but voted Remain in the referendum, said the deal being negotiated with the EU "will be a terrible mistake".
Arguing Britain was "on the brink of the greatest crisis" since World War Two, he said what was on offer wasn't "anything like what was promised".
Downing Street thanked him for his work but ruled out another referendum.
Jo Johnson voted to remain in the EU in the 2016 referendum while his brother Boris, who quit as foreign secretary in July, was a leading Brexiteer.
His brother praised his decision, saying they were "united in dismay" at the PM's handling of the negotiations.
Cabinet ministers have been invited this week to read the UK's draft withdrawal deal with the EU. Theresa May has said the withdrawal deal is 95% done - but there is no agreement yet on how to guarantee no hard border in Northern Ireland.
On Friday the DUP, whose support Theresa May relies on for votes in the Commons, said they cannot support any deal which included the possibility that Northern Ireland would be treated differently from the rest of the UK.
Mr Johnson, the MP for Orpington in Greater London, said the choice being finalised was either:
- an agreement which would leave the UK "economically weakened with no say in the EU rules it must follow", or
- a "no-deal Brexit" which would "inflict untold damage on our nation".
He described this as "a failure of British statecraft unseen since the Suez crisis" but said even a no-deal Brexit "may well be better than the never-ending purgatory" being put forward by the prime minister.
But in a warning to his brother and fellow Brexiteers, he added: "Inflicting such serious economic and political harm on the country will leave an indelible impression of incompetence in the minds of the public."
The "democratic thing to do is to give the public the final say", he argued.
Analysis by BBC political editor Laura Kuenssberg
For some time, Jo Johnson has struggled with the unfolding reality of Brexit.
A well-respected and liked member of the government, he has decided that what was promised to people during the referendum campaign is now so different to what is on the table that he has quit the government instead.
He's not the first, nor the best-known minister to resign over Brexit. But to leave at this moment, right when Theresa May is trying to stitch together a final deal, could have a serious impact.
He added: "This would not be about re-running the 2016 referendum, but about asking people whether they want to go ahead with Brexit now that we know the deal that is actually available to us, whether we should leave without any deal at all or whether people on balance would rather stick with the deal we already have inside the European Union.
"Britain stands on the brink of the greatest crisis since the Second World War. My loyalty to the party is undimmed. I have never rebelled on any issue before now.
"But my duty to my constituents and our great nation has forced me to act."
In response, a Downing Street spokesman said: "The referendum in 2016 was the biggest democratic exercise in this country's history. We will not under any circumstances have a second referendum.
"The prime minister thanks Jo Johnson for his work in government."
Mr Johnson is the sixth minister in Theresa May's government to resign specifically over Brexit, following David Davis, Boris Johnson, Philip Lee, Steve Baker and Guto Bebb.
For Labour, Shadow Brexit Minister Jenny Chapman said Mrs May had "lost all authority and is incapable of negotiating a Brexit deal within her own party, let alone with the EU".
But asked in an interview with the German magazine Der Spiegel whether he would stop Brexit if he had the chance, Jeremy Corbyn replied: "We can't stop it, the referendum took place."
Lib Dem leader Sir Vince Cable, whose party supports calls for a "People's Vote" on the final deal, said: "We warmly welcome Jo Johnson's support of the campaign to give the people the final say on the deal and a chance to exit from Brexit.
"This is a fascinating situation in which Jo and his sister are united in opposing their brother Boris and his Brexit plans."
Brexiteer Tory MP Andrea Jenkyns tweeted that she did not agree with him about another referendum - but his intervention highlighted unease on both sides of the debate, with the PM's efforts to secure a deal.
And pro-Remain Conservative Anna Soubry supported his decision and said it was time for another referendum.
Huge respect for @JoJohnsonUK . It’s tough resigning from a Ministerial post, he’s done the right thing. Now is the time for people to stand up for what they believe in or we will sleepwalk to a #Brexit disaster. There is another way – it’s a @peoplesvote_uk— Anna Soubry MP (@Anna_Soubry) November 9, 2018
David Davis, who quit as Brexit Secretary over Mrs May's Chequers Brexit plan, tweeted:
Sorry to see @JoJohnsonUK resign. He's right that the Government's current proposals are "a travesty of Brexit" and represent a huge democratic deficit - out of Europe but run by Europe. However, a 2nd referendum is not the way forward and is not supported by the public.— David Davis (@DavidDavisMP) November 9, 2018