Mr Johnson quit as transport minister and minister for London in order to vote against the Brexit deal whenever it comes before Parliament and called for the public to be given a second referendum.

The senior Tory, who campaigned for Remain, follows his Leave-supporting brother Boris Johnson out of Mrs May’s government.

The former foreign secretary quit in July in protest at Mrs May’s handling of Brexit.

In a blog explaining his decision, Jo Johnson said: “It has become increasingly clear to me that the withdrawal agreement, which is being finalised in Brussels and Whitehall even as I write, will be a terrible mistake.

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“Indeed, the choice being presented to the British people is no choice at all.

“The first option is the one the Government is proposing: an agreement that will leave our country economically weakened, with no say in the EU rules it must follow and years of uncertainty for business.

“The second option is a no-deal Brexit that I know as a transport minister will inflict untold damage on our nation.

“To present the nation with a choice between two deeply unattractive outcomes, vassalage and chaos, is a failure of British statecraft on a scale unseen since the Suez crisis.”

He added: “Given that the reality of Brexit has turned out to be so far from what was once promised, the democratic thing to do is to give the public the final say.”

Boris Johnson backed his brother’s decision, saying: “We may not have agreed about Brexit but we are united in dismay at the intellectually and politically indefensible of the UK position.”

Jo Johnson acknowledged that the Brexit negotiations “have at least united us in fraternal dismay”.

He said that the terms of the Brexit deal being discussed with the EU would mean deciding key issues in the future relationship being put off while the UK is kept in a “boundless transitionary period”.

“This is a con on the British people: there is no evidence that the kind of Brexit that we’ve failed to negotiate while we are still members can be magically agreed once the UK has lost its seat at the table.”

He acknowledged that a no-deal Brexit could result in “Kent becoming the Lorry Park of England”, with real questions about guaranteeing supplies of food and medicines.

But even a no-deal Brexit “may well be better than the never-ending purgatory” that Mrs May’s plan would offer.

However, in a pointed message to his brother and other Brexiteers, he said: “Inflicting such serious economic and political harm on the country will leave an indelible impression of incompetence in the minds of the public.

“It cannot be what you wanted nor did the 2016 referendum provide any mandate for it.”

He said the public should be asked to confirm their decision to leave the EU and, if they choose to do that, whether to accept Mrs May’s plan or leave without a deal.

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’s resignation came just hours after Mrs May’s de facto deputy said he hoped a deal would result in a “new dynamic” at Westminster, with MPs rallying behind the Prime Minister.

David Lidington said a UK-EU deal would involve “compromises, give and take on all sides” but when faced with “product on the table” in the form of an agreement backed by all 28 governments there could be a shift in attitude at Westminster.

“I hope and I believe that we can secure that majority in Parliament for the agreement,” he said.

Hopes of an imminent breakthrough in the Brexit negotiations have so far failed to be met.

But both Mr Lidington and Ireland’s Leo Varadkar suggested a deal could be reached between the UK and EU in the coming weeks.

The Taoiseach said: “A successful outcome is not guaranteed but I think it is possible in the next couple of weeks.”

But the fragile alliance keeping Mrs May in power has been strained as the Democratic Unionist Party railed against measures it fears will create a border down the Irish Sea between Northern Ireland and the rest of the UK.

DUP leader Arlene Foster said the Prime Minister appeared “wedded to the idea of a border down the Irish Sea” despite Downing Street’s repeated assurances to the contrary.

Minister Jo Johnson resigns over Brexit and backs second referendum