Video report by ITV News Political Correspondent Carl Dinnen
Theresa May has pledged to protect the "constitutional integrity" of the UK in a bid to break a deadlock in Brexit talks over the Irish border.
During Prime Minister's Questions, she told MPs: "We will ensure there is no hard border between Northern Ireland and the Republic."
It comes after Mrs May was forced to pull out of a potential deal with the EU after meeting fierce resistance from Democratic Unionists about a proposal for there to be continued "regulatory alignment".
The party said it will not accept any Brexit deal that "separates" Northern Ireland from the rest of the UK with a hard border.
Earlier, the Prime Minister spoke by telephone to DUP leader Arlene Foster, who has said the party got a "big shock" when shown the proposed text of Mrs May's offer to the EU on Monday.
"We that knew we couldn't sign up to anything that was in the text that would allow a border to develop in the Irish sea," she added.
Mrs May had expected to speak to Mrs Foster on Tuesday but the call did not take place.
ITV News Political Editor Robert Peston said the phone call between Mrs May and the DUP leader was a "gesture" and no progress had been made over the border issue.
He added that the government's hopes to conclude a deal by Sunday appeared to be "ambitious".
Commenting on the phone call between Mrs May and Mrs Foster, Irish Taoiseach Leo Varadkar said the DUP did not represent everybody in Northern Ireland.
"We need to bear in mind that there are a lot of different voices in Northern Ireland; we need to listen to them all, and all parties in Northern Ireland, not just one."
He told the Dail parliament in Dublin that it was the ambition and wish of his Government to begin phase two of the UK-EU talks to address post-Brexit trade and acknowledged it was in the Republic's own interest, but added that "if it is not possible to move to phase two next week then we can pick it up in the new year".
Mr Varadkar continued that he stood by the text of a draft deal "agreed" on Monday.
ITV News Political Editor Robert Peston on whether or not a decision between the UK and Ireland is near, and why the Cabinet is divided over a trade deal.
Mrs May spoke to Mr Varadkar by phone on Wednesday afternoon, and according to Number 10 "both agreed about the paramount importance of no hard border between Ireland and Northern Ireland", with the Prime Minister recognising "the significance of this issue to the people of Northern Ireland and Ireland" and that the issue remains a "priority... to resolve".
The spokesperson continued that the Government and the EU are "working hard to find a specific solution to the unique circumstances in Northern Ireland that respects the integrity of the UK, the European Union and the Belfast Agreement.
"She added that we are committed to moving together to achieve a positive result on this as well as restoring devolved Government to Northern Ireland.
"Both leaders looked forward to continuing relations as close neighbours and allies as the negotiations progress."
A spokesperson for Mr Varadkar continued that the pair "agreed to speak again over the coming days", and in Wednesday's conversation they had taken "stock of developments since Monday", with "the Taoiseach reiterating the firm Irish position regarding the text as outlined by him on Monday".
With Brexit dominating PMQs, Labour leader Jeremy Corbyn labelled the Government's approach a "shambles", adding it was "not fit for the future".
"If they can't negotiate a good deal wouldn't it be better if they just got out of the way," he added.
Mr Corbyn also said there were "1.5 billion reasons why" Mrs May should have shared details of the Irish border proposals with the DUP, in a nod to her minority government's Westminster pact.
The PM was heckled as she said "very good progress" was being made in the Brexit talks before insisting the Government believed it could get the "right" deal.
Mrs May also faced further heckling as she noted European Commission president Jean-Claude Juncker had said there were "still a couple of things" being negotiated.
The Prime Minister also insisted the only "hard border" is down the middle of the Labour Party.
"Half the Labour Party wants to stay in the single market, half the Labour Party wants to leave the single market," she said.
Meanwhile, Mrs May was warned in the Commons that no Brexit deal would be "reckless and dangerous", as "utterly ruthless" dissident republicans would target border officials between Northern Ireland and the Republic of Ireland.
Independent MP Lady Hermon (North Down) said a hard border would inevitably exist between Northern Ireland and the Republic if the UK and EU failed to agree a deal.
She added the UK Government had a "moral" responsibility to take care of all officials, including those from HM Revenue and Customs, Police Service of Northern Ireland and UK Border Force.
Meanwhile, Brexit Secretary David Davis has been accused of misleading Parliament after admitting the Government has made no formal assessment of the likely impact of Britain leaving the European Union.
Mr Davis said there has been "no systematic impact assessment".
He added: "I'm not a fan of economic models because they have all proven wrong. When you have a paradigm change - as happened in 2008 with the financial crisis - all the models were wrong."
"Similarly, what we are dealing with here in every outcome - whether it is a free trade agreement, whether it is a WTO outcome or whether it is something between that on the spectrum - it is a paradigm change."
On Wednesday, the Chancellor also revealed that the Cabinet had not yet had a full discussion of what should be the Government's preferred "end state position" for the UK post-Brexit.
Speaking to the Commons Treasury Committee, Philip Hammond said: "The Cabinet has had general discussions about our Brexit negotiations, but we haven't had a specific mandating of an end-state position.
"That is something that will be done first in the sub-committee constituted to deal with this issue, and logically that will happen once we have confirmation that we have reached 'sufficient progress' and are going to begin the phase two process with the European Union.
"We are not yet at that stage and it would have been premature to have that discussion before we reach that stage."
The EU insists trade negotiations can begin only after "sufficient progress" has been made on the three key divorce issues of the Irish border, citizens' rights, and the UK's exit bill.
Mrs May and European Commission President Jean-Claude Juncker hope to reconvene later this week for more talks ahead of the summit of the European Council on December 14.
Foreign Secretary Boris Johnson has called on the EU to give an immediate green light to talks on post-Brexit trade relations, rather than insisting on waiting for a breakthrough over the Irish border.
"We will come up with a solution, but the important thing is that that solution can only be discovered in the context of discussions on the end-state of the UK's relations with the rest of the EU.
He added: "What I would say is that the best way to sort it out is to get on to the second phase of the negotiations, where all these difficult issues can be properly teased out, thrashed out and solved."