* "Sufficient progress" agreed in talks, Commission says
* Brexit talks to move onto trade and transition
* Hopes rise of an orderly Brexit, sterling rises
* Pressure eases on PM May after fears of her imminent fall (Recasts, adds details)
BRUSSELS, Dec 8 (Reuters) - Britain and the European Union struck a divorce deal on Friday that paves the way for arduous trade talks, easing immediate pressure on Prime Minister Theresa May and boosting hopes of an orderly Brexit.
May rushed to Brussels before dawn to seal European Commission agreement that "sufficient progress" had been made to begin talks about trade and a two-year Brexit transition period that will start when Britain leaves the EU on March 29, 2019.
Negotiators in London, Brussels and Dublin worked through the night before breaking an impasse over the status of the Irish border, the last major obstacle to the opening of trade talks which EU leaders are due to bless at summit on Dec. 14-15.
Speaking before sunrise at the EU's executive headquarters in Brussels after a hurried flight on a Royal Air Force plane, May said opening up trade talks would bring certainty for citizens and businesses about Britain's future after quitting the EU.
"The most difficult challenge is still ahead," European Council President Donald Tusk cautioned. "We all know that breaking up is hard. But breaking up and building a new relationship is much harder."
May, looking weary after just a couple of hours sleep, spoke after European Commission President Jean-Claude Juncker announced the breakthrough first in English and then in German and French.
The move to trade talks 18 months after the United Kingdom's shock vote to exit the EU allayed some fears of a disorderly Brexit that could disrupt trade between the world's biggest trading bloc and its sixth-largest national economy.
Sterling climbed to a six-month high against the euro on Friday before it fell back around midday to sit broadly flat, with one euro worth 87.4 pence, while bond yields across the euro zone rose. Against the U.S. dollar the pound also weakened.
Facing 27 other members of the bloc, May largely conceded to the EU on structure, timetable and substance of the negotiations.
Moving to talks about trade and a Brexit transition was crucial for May's own future after her premiership was thrown into doubt when she lost the ruling Conservative Party its majority in a snap election in June, unwisely called.
"I very much welcome the prospect of moving ahead," said May, a 61-year-old Anglican vicar's daughter who herself voted to stay in the EU in a referendum in June 2016 but has repeatedly insisted Britain will make a success of Brexit.
One senior British banker said the deal signaled that May would stay in power for now and that Britain was heading towards a much closer post-Brexit relationship with the EU than many had feared.
Draft guidelines showed the transition period, which would start on March 29, 2019, would last around two years. During that time, Britain will remain part of the customs union and single market but will no longer take part in EU institutions or have a vote.
It will also still be subject to EU law.
Pro-Brexit Conservative lawmakers rallied around her after the overnight deal, a possible signal that the party - which has been split over EU membership for generations - was not preparing to ditch her immediately despite the June election fiasco.
British Foreign Secretary Boris Johnson, who spearheaded the Brexit campaign, congratulated May, adding that Britain would now take back control of its laws, money and borders.
Supporters of a radical Brexit were tougher.
Brexit campaigner Nigel Farage struck a jarring note saying it was extraordinary a British premier had conceded so much in the middle of the night, agreeing to all the demands of Juncker, Tusk and EU negotiator Michel Barnier.
"The British prime minister has to fly through the middle of the night to go and meet three unelected people, who condescendingly say: 'Now jolly well done May, you've met every single one of our demands, thank you very much, we can now move on to the next stage'."
The EU had insisted it would only move to trade talks if there was enough progress on three key issues: the money Britain must pay to the EU; rights for EU citizens in Britain and British citizens in the EU; and how to avoid a hard border with Ireland.
"I believe we have now made the breakthrough we needed," Juncker said.
The EU's chief Brexit negotiator Michel Barnier said it was not possible to put a concrete figure on the amount of money Britain will have to pay. Britain has said the divorce bill will cost it between 35 and 39 billion pounds.
On citizens rights, London and Brussels agreed to offer equal treatment on social security, health care, employment and education and that Britain will enable its judges to ask the European Court of Justice to weigh in when necessary for eight years after Brexit.
But the crucial breakthrough was on the future of the 310-mile (500 kms) UK-EU land border on the island of Ireland. The Northern Irish party which props up May's minority government vetoed a draft deal on Monday.
May worked through most of the night, grabbing just a couple of hours sleep, as she worked the phones from Downing Street to secure agreement from Dublin, Brussels and the Democratic Unionist Party for her deal on the border.
They agreed to avoid a hard border which might upset the peace established after decades of violence, but said the details would be agreed as part of talks about the future relationship, according to a 15-page negotiators report.
In the text, Britain agreed that should London and Brussels fail to agree a final Brexit deal, the United Kingdom will maintain "full alignment" with those rules of the internal market and customs union that help protect north-south cooperation in Ireland.
"In all circumstances, the United Kingdom will continue to ensure the same unfettered access for Northern Ireland's businesses to the whole of the United Kingdom internal market," it said. .
The Democratic Unionist Party gave only a conditional endorsement of the new terms, four days after 11th-hour objections from Belfast scuppered May's attempt to sign off on an accord over the Irish border.
"We cautioned the Prime Minister about proceeding with this agreement in its present form given the issues which still need to be resolved," Democratic Unionist Party (DUP) leader Arlene Foster said.
"Nothing is agreed until everything is agreed and how we vote on the final deal will depend on its contents."
(Writing by Guy Faulconbridge and Kate Holton in London; Additional reporting by Alistair Smout, William James, Costas Pitas and Andrew MacAskill in London, Padraic Halpin in Dublin, and; Editing by Richard Balmforth)