Prime Minister Theresa May announced Hunt's new role in a tweet, adding that the queen had approved the change.
Hours later UK Foreign Minister Boris Johnson resigned over the political crisis.
Johnson said in his letter that May's plan to keep close economic ties with the bloc means the U.K.is heading for a "semi-Brexit" that would leave Britain with the "status of colony" within the EU.
The resignation on Monday of both Johnson and a second Brexiteer cabinet minister, Brexit secretary David Davis, marks the end of an experiment: could May chart a course towards an agreement that satisfied, or at least appeased, the Brexiteers, Remainers, pragmatists and miscellaneous in her party?
This is within hours of David Davis quitting as the Brexit secretary and being replaced by Dominic Raab, which is causing many to believe that a general election is around the corner. He said "the Brexit dream is dying, suffocated by needless self doubt".
Labour leader Jeremy Corbyn said Mrs May should make way for a Labour administration if her Government can not get its act together quickly.
May's plan seeks to keep the United Kingdom and the European Union in a free-trade zone for goods, and commits Britain to maintaining the same rules as the bloc for goods and agricultural products.
May imposed on her cabinet on Friday.
She told her critics the alternative to the party coming together could be a left wing Labour administration, with Jeremy Corbyn as prime minister.
Yet he and Johnson have shared a mutual affinity for one another, so his departure may put the already tenuous U.S. -U.K. relationship on even shakier ground, especially if May enters into conversations with Trump from a weakened position. They also argued that the proposals breach several of the "red lines" that the government had set out, including a commitment to leave the EU's tariff-free customs union.
Whether Theresa May or another is at the helm in the UK, EU diplomats say they need a leader who has to power to make a deal in Brussels and see it through back home.
The UK and the European Union have been negotiating Brexit terms for more than a year now and have been hoping to agree broad aims for their future relationship in October.
Asked if Brexiteers needed to put the PM's future to a vote of the Conservative Party, he replied "it may well come to that".
"Of course this is a complex area of judgement and it is possible that you are right and I am wrong".
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