May’s statement inevitably triggered the starting gun on a two-month contest to replace her.
May will step down as Conservative leader on June 7 but stay on as prime minister until party members have chosen her successor, which will happen by July 20.
Britain’s EU departure date is currently fixed for October 31, although any new leader could ask for a further delay.
Conservative Party leadership contests are typically bloodthirsty affairs, with plot twists and betrayals.
Former foreign secretary Boris Johnson is the bookmakers’ odds-on favourite, ahead of former Brexit secretary Dominic Raab.
Both have embraced the prospect of a no-deal Brexit.
But whoever takes over from May will like her face the same razor-thin majority in parliament, an EU with no intention of changing the divorce offer that MPs have rejected three times, and proponents both for and against Brexit who will never compromise.
Main opposition Labour leader Jeremy Corbyn said whoever wins the contest should call an immediate general election.
That too would be a risky move, with the newly-formed, single-issue Brexit Party set to triumph in the European Parliament elections when the results become clear on Monday.