Britain is likely to enter arduous talks on its exit from the European Union without a deal to keep Prime Minister Theresa May in power as negotiations with a Northern Irish "kingmaker" party grind into a second week.
The Brexit Secretary said the talks would take place that week, but not necessarily on the Monday due to the clash with the Queen's Speech at Westminster.
Barnier, the EU's chief negotiator, said there was no desire to punish the United Kingdom but "its accounts must be settled".
"Michel Barnier, the European Commission's Chief Negotiator, and David Davis, Secretary of State for Exiting the European Union, agreed today to launch Article 50 negotiations on Monday, 19 June", writes the common statement of the European Commission and the European Council, after an agreement with Barnier's counterpart was reached.
"There is a unity of goal among people in the United Kingdom", May said following a meeting with French President Emmanuel Macron in Paris.
But speaking in Strasbourg, Guy Verhofstadt, the former Belgian Prime Minister, who is now responsible coordinating the European Union parliament's Brexit stance, suggested there would be a price to pay should Britain wish to remain in the bloc.
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But a newly appointed junior Brexit minister, Steve Baker, told Reuters: "I don't foresee any change".
Hundreds of protesters chanting "we want justice" stormed a local town hall on Friday afternoon and May had to leave a meeting with residents under heavy police guard.
Barnier this week acknowledged "sensitivity" in London at European Union suggestions that Britain might owe it some 60 billion euros in 2019 and said sorting out the issue soon would help a trade deal: "I would like to very quickly play down this question, and find concrete, pragmatic and just solutions", he said on Monday. Like Alice in Wonderland, not all doors are the same. "10 months later find that there was no real majority for that?" the official said.
"We would restore faith in politics if we could show that this parliament can at least function in presenting a view in the national interest which would command a majority on a cross-party basis", said veteran pro-European Conservative lawmaker Ken Clarke.
If no trade deal has been reached by this point, UK-EU relations could be governed by a "transitional arrangement" for some months or years. The DUP is one of the most socially conservative parties in Europe, having sought to maintain some of its strictest limits on abortion and consistently opposed gay marriage.
However, putting the pro-British unionist DUP in a position of influence in London could also undermine the British government's ability, enshrined in a 1998 peace agreement, to function as an impartial broker between Northern Ireland's unionists and its Catholic Irish nationalists.
Following the talks, Sinn Fein president Gerry Adams said the party would not stand for any Conservative-DUP arrangement which compromised British government impartiality in Northern Ireland.