But a deal was worked out overnight, and Mr. Juncker said the commission was satisfied that “sufficient progress” has been made in three areas:
• The question of the border between Ireland and Northern Ireland. A spokesman for the Irish government said, “This is a very good day for Ireland, North and South.”
• A financial settlement, frequently referred to as a divorce bill. Mrs. May had already roughly doubled her original financial offer to the bloc, to pay for commitments made while Britain was a member.
• The rights of British and European Union citizens. There are three million citizens of the bloc in Britain and one million Britons in the union, and their fate had emerged as a sticking point.
While negotiators managed to finesse the Irish border issue to reach this agreement, the matter seemed far from settled. It will now go to trade negotiators, and Prime Minister Leo Varadkar of Ireland noted approvingly that there was now a “backstop arrangement,” in case they do not resolve the issue.
Under that deliberately ambiguous formulation, Northern Ireland and perhaps all of the United Kingdom would maintain “full alignment” with European rules as needed to “support North-South cooperation, the all-island economy and the protection of the 1998 Agreement” that ended the Troubles in the North.
The haziness surrounding the arrangement was cause for concern for Ms. Foster and the Democratic Unionist Party, who are determined above all to avoid a situation in which the rules governing Northern Ireland diverge from those for the rest of the United Kingdom. That direction, they fear, would ultimately lead to reunification with the South.
So, while welcoming the idea there would be no “red line,” or border, running through the Irish Sea, Ms. Foster said in a statement, “We cautioned the prime minister about proceeding with this agreement in its present form, given the issues which still need to be resolved and the views expressed to us by many of her own party colleagues.”
But that snag, should it develop at all, lies in the future, while Friday was portrayed as a day for celebration, however muted by recognition of the hard road ahead.
“This is a difficult negotiation but we have now made a first breakthrough,” Mr. Juncker said in a statement. “I am satisfied with the fair deal we have reached with the United Kingdom. If the 27 member states agree with our assessment, the European Commission and our chief negotiator Michel Barnier stand ready to begin work on the second phase of the negotiations immediately.”
The heads of the member states will meet next week and are expected to confirm the deal next Friday.
“This government will continue to govern in the interests of the whole community in Northern Ireland and uphold the agreements that have underpinned the huge progress that has been made over the past two decades,” Mrs. May said in a statement on the British government’s website.
With the 2019 deadline fast approaching, Mrs. May has been under growing pressure to make progress. Opponents have criticized the way she has conducted the negotiations, and British businesses have been increasingly anxious to know what rules will apply after British withdrawal, known as Brexit.
Mrs. May, who lost her parliamentary majority in elections earlier this year, has been assailed by numerous problems at home, where her cabinet is deeply divided over Brexit policy and two ministers were recently forced to quit over separate issues.
Although supporters of Brexit once insisted that Britain held all the cards in the withdrawal negotiations, it has been Mrs. May who has made nearly all the concessions.
Assuming that European Union leaders agree at their summit meeting in Brussels to proceed, detailed trade negotiations will begin soon, probably early in the new year. There will also be talks on a transition period, which Mrs. May wants to run for two years, during which very little will change as business gets ready for a new set of rules.
Time is short. In March, Mrs. May triggered Article 50 of the European Union’s treaty, which lays down a two-year timetable to forge an exit agreement. That can only be extended with the agreement of all member nations.
A trade and transition agreement will have to be concluded well before March 2019 — probably by the fall of 2018 — in order to provide time for it to be ratified by member nations and by the European Parliament. Most trade experts believe that it will only be possible to agree on an outline trade deal next year, and that negotiations will be extended into the transition period.
The tight time span is not Mrs. May’s only problem. Her cabinet has yet to agree on its ultimate objectives for Brexit, particularly the extent to which Britain would continue to adopt the standards and rules of the European Union, its neighbor and biggest trading partner.
“While being satisfied with this agreement, which is obviously the personal success of Prime Minister Theresa May, let us remember that the most difficult challenge is still ahead,” said Donald Tusk, the president of the European Council. “We all know that breaking up is hard, but breaking up and building a new relation is much harder.”
Mrs. May’s colleagues are split between those who want to remain close to the bloc, to minimize the disruption to trade, and hard-line supporters of Brexit, like the foreign secretary, Boris Johnson. He wants Britain to be able to adopt its own rules and regulations and to be free to strike new free trade deals with non-European nations around the globe.
In a speech in September in Florence, Mrs. May said that while Britain would leave the customs union and single market, she wanted a much deeper free trade agreement than the one the European Union negotiated with Canada. Officials with the bloc insist that Britain cannot be outside the bloc’s main economic structures and still receive the same type of market access as those on the inside.
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