Nearly 16 months on, she still refuses to say whether she has since changed her mind. In an interview with LBC radio presenter (and CNN contributor) Iain Dale, the PM said that she would not "answer hypothetical questions."
During the referendum campaign she gave just one public speech and limited her campaigning to photo opportunities for local media. (Photos of the woman who is now Britain's Prime Minister holding up an "I'm IN" sign are available on social media, but require some digging to unearth).
This low profile enabled her to win the Conservative Party leadership contest that followed David Cameron's resignation
after his referendum defeat. She was the only candidate then acceptable to both the party's "Leave" and "Remain" wings, and appointed a Cabinet balanced between both.
But she still needed to reassure the euroskeptics: that was the purpose of her infamous soundbite
: "Brexit means Brexit -- and we will make a success of it." I may have supported remaining in the EU then, she meant, but don't worry, I'm on your side now.
Pushed by her Brexit-supporting adviser Nick Timothy, she set Britain up for adversarial negotiations with Brussels.
The business community's preferred outcome, a so-called "soft" Brexit, where the UK left the political structures of the EU, but stayed in economic ones (known as the single market and customs union), was dismissed.
Moderate Cabinet ministers were forced to toe the euroskeptic line. During the campaign for the snap election May called this June, nobody even thought of questioning her pro-Brexit credentials.
Leave voters trusted her: research from the British Election Study shows that the Conservatives picked up five percentage points
of support from Leave voters.
Unfortunately for the Tories, her hard line drove Remain-supporting Conservatives away, and her party lost its majority in the House of Commons.
She has since clung on to office, but as the captive, rather than leader of her Cabinet.
Yet now that she must listen to it, it is not so much balanced as it is divided. She's too weak to lead her ministers, but they're too split to make her follow.
Hence the devastating impact of Iain Dale's question about how she would vote if there were another referendum on EU membership. If she had said that she would now vote to leave, she would have to explain why she changed her mind.
But if she said she would vote to remain, she would confirm growing Brexiteer fears that she -- along with some of her key Cabinet ministers -- didn't really support her government's main policy.
Indeed, her deputy Damian Green (a sincere and passionate pro-European) admitted he'd vote "remain" again and thought that Britain would be "better off" staying in, but that he had a democratic duty to carry out the instructions of the British people.
All of this leaves Britain in something of a bind.
Remain-supporting members of the Cabinet know that the best hope they have of mitigating disaster is to work from the inside, under the cover of a weak Prime Minister.
If they brought the government down, the Conservative Party's grassroots would pick the most anti-European leader possible. At best, this would waste valuable time (there are only 18 months left to negotiate); at worst, it would cause the negotiations to collapse.
Leave supporters see advantage in having former Remain supporters involved so they can be blamed for the costs imposed when Brexit goes wrong. But they are starting to ask how long they can continue to support a government whose leading figures think its main policy is a disaster to be mitigated, not an achievement to celebrate.
The stability of May's government is based on the willing suspension of disbelief: Leave supporters and former-Remain supporters pretending they mean the same thing when they say they want the "best Brexit deal."
But when leavers believe the best deal can be obtained by threatening to walk away from Europe without a deal, while the once remain supporting finance minister warns that "no deal" could mean planes would be prevented from flying between Britain and the EU, that suspension is called into question.
It is becoming clear that the government assembled after the June election is not capable of carrying out the revolutionary Brexit that hard line Brexiteers, most of the Tory party's members, and a significant minority of the British public want.
These Brexiteers see their government willingly capitulating to the EU's "intransigence." They fear the UK will end up like Norway: forced to accept EU rules without having any say in making them.
Dale's question to the Prime Minister exposed this fundamental fact. If the hardliners want their kind of Brexit, they will have to take back control of the process. Whether the Conservative Party could survive such a Bolshevik Brexit is another matter altogether. But it's becoming increasingly clear that the best May's government could achieve would be a fudged Brexit.