A joint statement said: "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 June 19".
While the DUP are deeply eurosceptic, they have balked at some of the practical implications of a so-called hard Brexit - including a potential loss of a "frictionless border" with the Republic of Ireland - and talks will touch on efforts to minimize the potential damage to Northern Ireland.
"But this is a decision for the British parliament and the British people on whether they want this door to be wide open again or whether, after making such a decision, they say, 'Let's consider how we can use some portions of this openness, but others perhaps not."' Gabriel said.
"I agree. But like Alice in Wonderland, not all the doors are the same. The intent is to ensure that we have the stability of Government in the national interest".
May's botched election gamble has left her so weakened that her Brexit strategy is the subject of public debate inside her party, with two former prime ministers calling on her to soften her European Union exit approach. The Conservatives are considering an arrangement in which the Northern Ireland party backs May on the budget and her confidence motions.
May is under pressure to take on a more cross-party approach to the negotiations surrounding Brexit.
And as leaders welcomed the new tone in London and talk of a "softer Brexit" that may be less disruptive than May's clean break with the single market and customs union, officials from at least some governments saw compromise on the British bill.
Even the idea of an alliance is complicated, however.
"Any deal which undercuts in any way the process here or the Good Friday Agreement is one which has to be opposed", he said.
DUP leader Arlene Foster, who met Mrs May in No 10 on Tuesday, is understood to have returned to Northern Ireland leaving her deputy Nigel Dodds to represent the party at Thursday's meeting.
Sinn Fein believes that it could win that referendum because of concerns in Northern Ireland about Brexit.
The stakes for May are high.
The partnership between the DUP and the Conservative (and Unionist) Party in the British government might in theory make Irish nationalists and republicans more willing to concede (Gerry Adams, the leader of Sinn Fein, has hinted that he might be prepared to make concessions on Foster's return). If that happens, Corbyn will demand a chance to try to form a government by uniting progressive factors in the House of Commons.
"And I confirmed to President Macron that the timetable for the Brexit negotiation remains on course and will begin next week".