BREXIT talks broke up in Brussels without a deal, after a proposed solution for the Irish border met fierce resistance from Arlene Foster's Democratic Unionist Party.
Discussions are set to resume later this week, with both Prime Minister Theresa May and European Commission president Jean-Claude Juncker declaring themselves "confident" that a solution can be found in time for a key summit of the European Council on December 14.
Mrs May began her lunchtime meeting with the Commission president with hopes high that Brussels would be able to declare that sufficient progress had been made on the so-called "divorce issues" in order to allow the leaders of the remaining 27 EU nations to give the green light for trade talks to begin next week.
But reports that Mrs May was on the verge of agreeing a deal on "regulatory alignment" between Northern Ireland the Republic led the DUP to warn it would not back any agreement which threatened the territorial integrity of the UK.
Mr Juncker said the meeting was "friendly and constructive".
He went on: "I have to say that she's a tough negotiator, and not an easy one, and she's defending the point of view of Britain with all the energy we know she has, and this is the same on the side of the European Union.
"Despite our best efforts and significant progress we and our teams have made over the past days on the three main withdrawal issues, it was not possible to reach a complete agreement today.
"We now have a common understanding on most relevant issues, with just two or three open for discussion.
"These will require further consultation, further negotiation and further discussions.
"We stand ready to resume the negotiations with the United Kingdom here in Brussels later this week.
"But I have to say that we were narrowing our positions to a huge extent today, thanks to the British Prime Minister, thanks to the willingness of the European Commission to have a fair deal with Britain.
"I'm still confident that we can reach sufficient progress before the European Council of December 15.
"This is not a failure, this is the start of the very last round. I'm very confident that we will reach an agreement in the course of this week."
Mrs May said: "We have had a constructive meeting today. Both sides have been working hard in good faith.
"We have been negotiating hard. And a lot of progress has been made. And on many of the issues there is a common understanding.
"And it is clear, crucially, that we want to move forward together. But on a couple of issues some differences do remain which require further negotiation and consultation.
"And those will continue, but we will reconvene before the end of the week and I am also confident that we will conclude this positively."
DUP leader Arlene Foster spoke out after Ireland's deputy premier Simon Coveney said the Dublin Government's concerns over the post-Brexit border with Northern Ireland were set to be addressed fully.
Regulatory alignment could mean both Ireland and Northern Ireland following the same rules governing trade, to ensure that goods can continue to move freely across a "soft" border with no checks.
But critics say that it would effectively move the customs border between the UK and the Republic into the Irish Sea.
How to maintain a soft Irish border has emerged as the key sticking point in Brexit negotiations, after London indicated it was ready to up its offer on the so-called "divorce bill" to as much as £50 billion.
Speaking ahead of Mrs May's meeting with Mr Juncker, Mr Coveney told RTE Radio One he believed that the post-Brexit border would be "invisible" with "no barriers" and "will look very much like it looks today".
But Mrs Foster made clear the DUP would oppose the deal if it meant Northern Ireland being subjected to different rules from the rest of the UK.
Speaking at Stormont, she said: "We have been very clear. Northern Ireland must leave the EU on the same terms as the rest of the United Kingdom.
"We will not accept any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the United Kingdom.
"The economic and constitutional integrity of the United Kingdom will not be compromised in any way."
The DUP, which props up Mrs May's minority Government in the House of Commons, has previously warned it could withdraw its support in Westminster if a deal is proposed which threatens the integrity of the United Kingdom.
Mrs Foster's intervention disrupted the choreography of events in Brussels, as the PM broke off from talks for urgent telephone discussions with the DUP leader.