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6 year oldHealth Secretary Jeremy Hunt has been named as the new foreign secretary after Boris Johnson quit, accusing Mrs May of pursuing a "semi-Brexit".
His departure followed that of Brexit Secretary David Davis and several junior figures.
Culture Secretary Matt Hancock replaces Mr Hunt as health secretary.
In his first comments as foreign secretary, Mr Hunt said he would be standing "four square" behind the prime minister "so that we can get through an agreement with the European Union based on what was agreed by the Cabinet last week at Chequers".
"This is a time when the world is looking at us as a country, wondering what type of country we are going to be in a post-Brexit world.
"What I want to say to them is Britain is going to be a dependable ally, a country that stands up for the values that matter to the people of this country, and will be a strong confident voice in the world."
Mr Johnson claimed in his resignation letter that Mrs May's proposals for post-Brexit trade would leave Britain a "colony" of the EU.
Mrs May said she was "sorry - and a little surprised" by Mr Johnson's move after his apparent support on Friday, when ministers held talks at Chequers.
She said the deal agreed by the cabinet after their "productive discussions" would "honour the result of the referendum" and allow the UK to "take back control of our borders, our law and our money".
The prime minister earlier faced down backbench critics at a meeting of the 1922 committee, amid rumours they were close to getting the 48 signatures needed to trigger a no-confidence vote that could have removed her as prime minister.
Jeremy Hunt, who has been health secretary for the past six years and recently secured a £20bn boost in funding for the NHS, was a Remain campaigner in the 2016 EU referendum.
He has since said he is a convert to the Brexit cause, but his appointment will be seen as balancing out the promotion of staunch Leave supporter Dominic Raab to the cabinet as Brexit secretary.
Both Mr Johnson and Mr Davis had signed up on Friday to Mrs May's blueprint for Brexit - but 48 hours later, first Mr Davis and then Mr Johnson said they could not commit themselves to promoting the plans as they did not believe in them.
Brexit minister Steve Baker and two ministerial aides also resigned.
In a scathing resignation letter to the PM, Mr Johnson said that, under her leadership, the dream of an outward-looking, global Britain after Brexit was "dying, suffocated by needless self-doubt".
In her reply Mrs May said she was "sorry - and a little surprised", at his decision but that if he could not support the government's position, "it is right that you should step down".
After his appointment as Health and Social Care Secretary was announced by Downing Street, Mr Hancock said: "Really looking forward to joining @DHSCgovuk at such an important time for our great NHS. I can't wait to get started."
Attorney General Jeremy Wright is the new culture secretary, with backbencher Geoffrey Cox replacing him as attorney general.
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