"His replacement will be announced shortly".
He added that he'd consider returning to Ukip to fight the Prime Minister's plans.
Al Jazeera's Laurence Lee, reporting from London, said Johnson's departure left the British government in crisis mode.
He told the programme: "What we are trying to do is not open revolt but we are trying to tell the prime minister and the Cabinet that we have got real concerns about where this is going".
In a two-page letter to Prime Minister Theresa May, the leading Brexit supporter said that while he initially accepted the government's proposal, it now "sticks in the throat".
In a remark addressed in particular to Mr Johnson, Mr Hague said: "Being a romantic on this issue is all very well but is of no practical use to the country".
"It delivers on the vote that people gave on Brexit, it delivers the fact that we will have an end to free movement, we will have an end to the jurisdiction of the European Court of Justice in the United Kingdom, we won't be sending vast contributions to the EU every year, we'll be out of the Common Agricultural Policy, out of the Common Fisheries Policy", she said.More news: "Fortnite" Season 5 Introduces New Vehicle & Desert Landscape
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The UK is set to leave the European Union on 29th March 2019, more than two years after the referendum that saw 51.9% vote to leave compared to 48.1% who wanted to stay.
Amid warnings from businesses that continued uncertainty is risking investment and jobs, and fears time is running out, her cabinet finally agreed on a plan last week.
Brexit Secretary Dominic Raab is insisting the much-anticipated Government white paper setting out its exit aims is practical and respects the referendum result while backing business.
The prime minister's official spokesman said she would fight any attempt to oust her as prime minister if the required 48 Tory MPs called for a contest. She's accused of betraying those who voted for Brexit, and her leadership might soon be challenged.
The Daily Telegraph said on Tuesday night that senior Brexiteers said that they have enough letters in hand to trigger a confidence vote and will submit them unless she hardens her Brexit plans.
The blueprint for leaving the European Union will be published later with ministers vowing to deliver a 'practical and principled Brexit'. He went on further still, remarkably claiming that the UK's inability to influence laws as they are made means "we are truly headed for the status of colony".
Just 30% of Britons think Theresa May is the best Conservative to lead Brexit negotiations, with 44% saying someone else within the party would be better suited, and 26% answered that they don't know.
May said she now wanted to advance the Brexit talks, saying she had briefed European Union leaders on her plan in recent days and received a "positive reaction". Steve Baker also resigned as a junior Brexit minister.