LONDON — Prime Minister Boris Johnson of Britain, in the boldest gamble of his high-wire political career, won backing on Tuesday to hold a general election on Dec. 12, throwing back to the British people the bedeviling issue of how, or even if, their country should leave the European Union.

The 438-20 vote in Parliament, which came after the opposition Labour Party dropped its resistance, provided the starting gun for one of the most momentous and unpredictable campaigns in post-World War II Britain, a six-week race that could forever alter Britain’s relationship to Europe and its place in the world.

Much will hinge on the sentiments of a fickle British public that is not just divided into warring camps but exhausted with the whole shambolic process and hoping for something, anything, finally to be decided — as long as it is not for the other side.

The motion to hold the election must still go to the House of Lords, where it could conceivably be held up, but that was unlikely.