Amber Rudd has been an impressive performer since her return to the cabinet last Friday, refreshed after seven months out since she resigned over the Windrush scandal. The new work and pensions secretary is a grown-up confident enough to answer media questions without sounding as if she is robotically reading out the official line.

Unfortunately for Theresa May, Rudd went dangerously off message on Radio 4’s Today programme this morning. She conceded that the Commons would not allow a no-deal exit next March if it votes down May’s Brexit deal. “It’s my view that parliament, the House of Commons, will stop no deal. There isn’t a majority in the Commons to allow that to take place,” she said.

Amber Rudd says Parliament will 'stop a no deal' Brexit from happening if Theresa May's agreement fails

This undermines May’s strategy of offering MPs a false choice between her deal and no deal, to scare both Labour and Tory backbenchers into supporting hers. For Eurosceptics, May adds a third option of “no Brexit”.

At Prime Minister’s Questions today, May said the alternatives to her deal were “either more uncertainty and division” or “no Brexit at all”. The first part is aimed at Labour MPs who fear a no-deal exit, the second at Tory Eurosceptics. But May’s messages are contradictory. She also said the UK would definitely leave the EU on 29 March. Which is not “no Brexit”.

Rudd’s candour exposes what almost everyone I talk to at Westminster says: if, as expected, the Commons rejects May’s deal next month, MPs will not march slowly like lemmings towards the cliff, and go over it next March. Whatever the parliamentary rules say about non-binding Commons votes, politics will trump them, and find a way to stop such a chaotic, economically damaging departure. Surely no prime minister could crash out of the EU like this without parliament’s support.

When she had the freedom to speak out, Rudd told ITV’s Peston on Sunday programme in September that a Final Say referendum was preferable to no deal. “I think a people’s vote could be the result of an impasse,” she said.

Rudd is one of many Tories who do not support a referendum today but might well do so if parliament is deadlocked. Another is Nicky Morgan, the former education secretary. Some ministers, who must oppose a Final Say vote for now, say privately they would switch if it became the best way to stop a no-deal exit.

Rudd’s comments today should provide food for thought for those MPs already committed to a referendum. The big moment they have been waiting for now comes into view, with the Commons vote on May’s deal only a few weeks away. MPs in all parties will use the debate to call for a referendum. For example, the Tory MP Sarah Wollaston, along with three other MPs who are clinicians, will table an amendment making approval of the withdrawal agreement contingent on the public giving their “informed consent” in a vote, as they would before surgery.

I believe a Final Say referendum is the right course, as The Independent has been arguing, and probably the only way out of the mess we’re in. But the Commons vote on the deal is not the right time to propose it.

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Although more than 100 MPs have backed a referendum, not all of them would vote for it at this stage. So it would be heavily defeated. That would allow May to try to take it off the agenda as MPs and the government decide what to do after the Commons rejects her deal, as almost everyone I talk to at Westminster expects. I can imagine the prime minister telling MPs: “This House has just considered a referendum. We have had an MPs’ vote and they overwhelmingly rejected a people’s vote. We had a people’s vote in 2016 and we must honour it.”

I understand why supporters of a referendum can’t wait to test the water. They do not want to miss the boat. They know that some cabinet ministers, believing May is doomed to lose the vote on the deal, are plotting a second vote after a few tweaks from Brussels.

But Rudd’s comments are a reminder that the Commons would have other options, including a referendum. At this point, the energy of its supporters would be best expended on voting down the deal. Much better to have a referendum vote after, not before, that. It would be a much more attractive option for many MPs then than it looks today.

A Commons majority for a Final Say vote can be built only with the support of the Labour opposition. Jeremy Corbyn is not convinced yet, but a growing number of Labour MPs, including frontbenchers, are. Proposing a referendum now would antagonise Corbyn, which would not be clever politics. Another reason why supporters should hold their fire.