Theresa May should resign after the crushing defeat of her Brexit deal with EU, Wales' first minister has said.
MPs rejected the prime minister's deal by 432 votes to 202 on Tuesday.
Mark Drakeford told AMs on Wednesday resignation was the "constitutionally proper course of action and she should face up to that and take it".
But Welsh Secretary Alun Cairns said Mrs May was "determined" to deliver on the 2016 referendum vote that the UK should leave the European Union.
AMs on Wednesday night backed a Plaid Cymru assembly motion against a no-deal Brexit - 37 versus 16.
Mr Drakeford said the prime minister has been defeated in Parliament "attempting to discharge the single most important responsibility that will ever fall to her".
However he rejected calls from his backbenches to swing behind a second Brexit referendum now.
Labour AM Lynne Neagle asked in the Senedd: "Would you agree with me that the only serious option now for those who want to put the people of Wales' first is a people's vote?"
Mr Drakeford replied: "I don't agree with that. I think we will come to the point when that may very well be the case.
"If we reach that point then going back to the people having a final say on this matter may have to be the way through this impasse that the House of Commons would find itself in if it wasn't able to... find a form of Brexit on which the House of Commons can agree."
Earlier, Mr Cairns said he remained "optimistic" that Parliament would not vote to stop the UK leaving the EU.
He told BBC Radio Wales: "I remain optimistic because I don't believe Parliament will vote in a way that will stop Brexit."
The prime minister plans cross-party talks if, as expected, she wins a no-confidence vote in her government later, tabled by Labour leader Jeremy Corbyn.
Mr Cairns said Mrs May would explain in the talks "the parameters of what Europe will accept".
"We in the Cabinet understand the risks of individual calls that some colleagues are making," he said.
"Some will be calling for one element of policy to be protected, not appreciating the consequences of what Europe will demand in another.
"So in these discussions the prime minister will be able to explain in greater detail 'well, if we demand one thing it potentially costs another thing'."
The vast majority of Wales' 40 MPs - 32 - voted against the Brexit deal. Only six Welsh Tory MPs supported it, with two of their colleagues opposed.
Welsh voters backed leave in the UK-wide EU referendum in 2016.
Brecon and Radnorshire MP Chris Davies, who backed Mrs May's deal, told BBC Wales: "My greatest fear is that Brexit will not go ahead and democracy will be ignored.
"I think that is the worst outcome for this country whether you are a Remainer or a Brexiteer," he said.
"I think the majority in the house has been for some time [for] a second referendum."
Labour leader Jeremy Corbyn is under huge pressure to back another referendum, with 71 Labour MPs, including 11 of the 28 in Wales, signing a letter to him calling for a new public vote.
They included Cardiff Central MP Jo Stevens, who said Mrs May should delay Brexit by extending the deadline under Article 50 and asking the public if they wanted to accept her deal or stay in the EU, which Ms Stevens said was "the best deal we can possibly get".
But Welsh Tory Brexiteer David Jones, who voted against the deal, said: "We've already had a referendum. The referendum was conclusive. Only in 2017 we had a general election when over 80% of the people who voted for parties who were committed to taking Britain out of the European Union.
"It makes a mockery of democracy if parties think they can keep putting the vote to the people until they get the vote that they want."
Mr Jones called for the prime minister to go back to the EU, saying: "I don't think there's any doubt now that the withdrawal agreement is dead."
Plaid Cymru is expected to back the no confidence vote but the party's Jonathan Edwards said it should not keeping backing further such votes, unless Tory rebels and the DUP vote against the government.
"What I will not entertain is supporting Jeremy Corbyn's plans for continuous no confidence motions," he said.
Despite predictions the prime minister would survive Labour's no confidence vote, a Tory MP who backed her EU deal warned anything was possible in the current febrile atmosphere at Westminster, and some in his party might just vote to depose their own government.
Carmarthen West and South Pembrokeshire's Simon Hart said: "I think the problem is that if half a dozen people do decide to do that, then Brexit's off.
"If there's an election, then not only will we have to delay Article 50, Jeremy Corbyn will probably get in - that's the end of Brexit.
"This project is over - finito."
In Cardiff Bay on Wednesday, Mr Drakeford met the Conservative leader in the assembly, Paul Davies, and Plaid Cymru leader Adam Price.
A Welsh Government source said it was to discuss the need to take the possibility of a no-deal Brexit "by accident" seriously and call on AMs to work across parties "to mitigate as far as within our power the consequences" of leaving the EU without a deal.
UKIP leader in the assembly Gareth Bennett complained he was excluded from the talks because he was "not part of the old boy's club in the cosy Cardiff Bay establishment".
"It's a slap in the face to the 17.4 million people across the UK who want the [UK] government to get on with the job [of Brexit], and another reason why devolution is failing the people of Wales.