Earlier, arriving at the meeting, the prime minister of the Netherlands, Mark Rutte, said that European leaders “should not underestimate” Mrs. May, adding that “she’s a formidable politician.” The German chancellor, Angela Merkel, met with Mrs. May for an aperitif before the leaders’ dinner. And the Belgian prime minister, Charles Michel, said he wanted to honor “the courage of Theresa May” during the talks so far.
The political thaw in part reflects some natural sympathy and growing respect for Mrs. May, who has battled on through months of infighting with her Conservative Party and frequent predictions of her political demise. As recently as Wednesday she suffered another reversal when some Conservative lawmakers supported a measure that will give Parliament a more significant say over the final terms of the withdrawal known as Brexit.
While European Union leaders still worry about Mrs. May’s prospects of surviving as prime minister, they seem to view her as a more pragmatic and reasonable interlocutor than some of the potential alternatives — like the foreign minister, Boris Johnson, who is roundly disliked in Brussels.
Mrs. May proved that by making a series of compromises during the first phase of negotiations, as the objectives of Brexit supporters clashed with the reality of dealing with a powerful bloc of 27 nations. Although Mr. Johnson once dismissed suggestions that Britain would offer huge sums of money to honor its outstanding commitments to the European Union, last Friday’s deal will cost London $47 billion to $52 billion, according to Mrs. May, and others think the actual figure will be even higher.
She also conceded a continuing, if time-limited, role for the European Court of Justice — an institution loathed by hard-line Brexit supporters — in adjudicating the rights of European Union citizens living in Britain.
The other divorce discussion involved offering assurances that there would be no hard border between Northern Ireland, which is part of the United Kingdom, and Ireland, which will remain in the European Union. That proved to be the toughest of the three separation challenges, but Mrs. May ultimately proved flexible in drafting an agreement, even if in reality it fudged rather than resolved the issues.
European leaders have noticed approvingly that Mrs. May has managed to make these concessions without provoking a rebellion from so-called Brexiteers in her fiercely divided cabinet. Ireland’s prime minister, Leo Varadkar, who played a leading role in the deal last Friday, said he had “absolute confidence” in Mrs. May, though he added that his priority was to turn what was agreed to last week into a “legally binding international agreement.”
On Friday the European Union leaders emphasized that there must be no backsliding on that deal, adding in a set of guidelines that “negotiations in the second phase can only progress as long as all commitments undertaken during the first phase are respected in full and translated faithfully into legal terms as quickly as possible.”
The early focus of those second-phase talks will be on a transitional program to cover the period immediately after March 2019 when Britain is to quit the bloc. During that time, which Mrs. May wants to limit to two years, very little will change in the relationship in order to give time to prepare for a potentially big shake-up in trade rules.
British businesses in particular are eager to have certainty that there will be a transition, rather than a sudden rupture — considered a “cliff edge” departure — from the bloc. To achieve that, Britain will have to accept all the rules of the European Union, including the remit of the European Court of Justice, during the transitional period.
But the bigger challenge is clinching a long-term agreement for a deep free-trade deal, given the constraints imposed by Mrs. May, who says Britain must leave the European Union’s single market and customs unions, which remove barriers to trade, and eliminate tariffs and customs checks.
Jean-Claude Juncker, president of the European Commission, said he expected formal discussions on future trade talks to start in March. While some British business leaders fret about the continuing uncertainty implied by such a delay, diplomats say they expect unofficial, exploratory exchanges over trade to begin earlier.
Mr. Rutte, the Dutch prime minister, said that it was now incumbent on Mrs. May to tell the other leaders what kind of trading relationship she wanted. He noted that she had been “holding her cards close to her heart,” adding that this had so far been “probably a wise negotiating tactic,” but one that would not hold for much longer.
Ms. Merkel reinforced that message, adding that it was not up to the European Union to lay out possible solutions to a problem created by Britain’s 2016 referendum decision to quit the bloc. “Britain has to tell us what it wants,” she said.
Phase 2 of the negotiations will be much harder from the European Union side, too. Thus far, it has had to hold together only on issues where there is consensus in European capitals. The question of a future trading relationship will involve forging agreement among 27 countries whose economic and political interests will diverge, depending upon the depth and character of their commercial and diplomatic ties to Britain.
Emmanuel Macron, France’s president, appealed to the 27 countries to stick together through the coming months, saying there must be “no separate agenda of bilateral talks,” reflecting, perhaps, a worry that Britain will try to divide the bloc.
Mr. Tusk, president of the European Council, said that he had “no doubts that the real test of our unity will be the second phase of the Brexit talks.”
As for the overall prospects, Mr. Tusk joined other leaders in predicting that the second phase would be more demanding. Asked whether Britain’s timetable for a Brexit agreement was attainable, Mr. Tusk described it as “still realistic and, of course, dramatically difficult.”