Where does this leave Britain?
Do not look to Brexiteers for answers.
Although they complain that the people have been betrayed,
they have still not explained how Britain could cut all ties with the EU while preserving trade links to what is by far Britain's largest market.
Mr Johnson did not even mention Ireland in his resignation letter this week.
It is as if Brexiteers have spent so many years in opposition attacking the EU that they are flummoxed by the idea of coming up with a workable plan.
While Mrs May at last faces up to the painful trade-offs that Brexit always required,
those who condemned her this week prefer to indulge their fantasies.
The EU could help—and has reason to.
It is reluctant to give Britain a bespoke deal,
for fear that its other restless members will angle for special treatment, too.
This is why Eurocrats solemnly vow that nothing must undermine the single market.
But if the Brexit negotiations fail, and Britain crashes out without any deal at all,
it would cause grave damage across Europe and beyond.
And in some areas the EU has an incentive to offer concessions.
The most glaring is security, where its hardline position is self-defeating.
Britain is one of Europe's two big military and intelligence powers.
Limiting its role in projects such as the Galileo geolocation system,
at a time when America is wavering on its NATO commitments and Russia is stirring up trouble, endangers all Europeans.
Bending rules such as freedom of movement is harder.
But the EU can help give Mrs May the cover she needs to sell the deal at home.
If she wants to replace free movement with a "mobility framework" that does much the same thing, let her.
If she wants market alignment on goods but not on services, so what?
And if Mrs May cannot win a Brexit vote?
Then the EU should be prepared to grant Britain more time, to avoid it crashing out without a deal.
To break the parliamentary impasse, Mrs May might have to go back to the people,
either with yet another election or even a second referendum,
setting out a concrete plan for Brexit rather than the vague, incompatible promises put before voters the last time round.
That Britain has at last set a course for a soft Brexit is welcome.
Getting there will be a very rough crossing indeed.