Hunt, a history lecturer, will resign next week as Labour MP for Stoke-on-Trent in central England as he prepares to take the prestigious post in the "coming months", he and the London museum said.
The announcement brings a fresh challenge for beleaguered Labour leader Jeremy Corbyn, who is struggling in opinion polls against Prime Minister Theresa May's Conservatives.
In his resignation letter, Hunt cited his "frustration" at the direction of his party under the veteran socialist, although he said he had "no desire to rock the boat".
In the June referendum on Britain's membership of the EU, he campaigned against Brexit -- but 69 percent of his constituents voted to leave the bloc.
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UKIP, a driving force behind Brexit under its former leader Nigel Farage, confirmed it would contest the forthcoming by-election.
"UKIP will be fighting hard to offer local people the committed representation they both need and deserve," said UKIP chairman Paul Oakden.
UKIP, which currently has only one MP, has been plagued by in-fighting since the vote to leave the EU, which was for decades its primary objective.
V&A Chairman Nicholas Coleridge said he had a "highly compelling mixture of experience across public life, the arts, history, education and academia, and knows our collections well".
As MP for an area known for its potteries, Hunt also helped save the Wedgwood Collection in 2014, which was gifted to the V&A and is on long-term loan to the Wedgwood Museum in Barlaston, Stoke-on-Trent.
Hunt said he was "delighted and honoured", adding: "I have loved the V&A since I was a boy, and today it is a global leader in its unrivalled collections, special exhibitions, academic research and visitor experience."