* Negotiators meet a year after Brexit referendum shock
* Davis wants "deep and special partnership" with EU
* May's election debacle leaves British plans unclear
* EU heartened by new president Macron's sweep in France (Updates with Davis, Barnier meeting)
BRUSSELS, June 19 (Reuters) - Brexit Secretary David Davis arrived in Brussels on Monday to launch talks he hoped would produce a "new, deep and special partnership" with the EU in the interest of Britons and all Europeans.
Beaming as he met the European Union's chief negotiator Michel Barnier at the EU executive's Berlaymont headquarters, the veteran campaigner for Britain to quit the bloc said he aimed for a "positive and constructive" tone in the talks, adding: "There is more that unites us than divides us."
Barnier, a former French minister, has voiced impatience in the past that Britain has taken nearly a year to open talks. Looking more sombre than his British counterpart, he said he hoped they could agree a format and timetable on Monday.
His priority, he said, was to clear up the uncertainties which last June's Brexit vote had created. He and Davis are due to give a joint news conference in the evening.
Almost a year to the day since Britons shocked themselves and their neighbours by voting on June 23 to cut loose from their main trading partner, and nearly three months since Prime Minister Theresa May locked them into a two-year countdown to Brexit in March 2019, almost nothing about the future is clear.
Even May's own immediate political survival is in doubt, 10 days after she lost her majority in an election.
Officials on both sides play down expectations for what can be achieved in one day. EU diplomats hope this first meeting, and a Brussels summit on Thursday and Friday where May will encounter - but not negotiate with - fellow EU leaders, can improve the atmosphere after some spiky exchanges.
Davis's agreement to Monday's agenda led some EU officials to believe that May's government may at last be coming around to Brussels' view of how negotiations should be run.
May's election debacle has revived feuding over Europe among Conservatives that her predecessor David Cameron hoped to end by calling the referendum and leaves EU leaders unclear on her plan for a "global Britain" which most of them regard as pure folly.
While "Brexiteers" like Davis have strongly backed May's proposed clean break with the single market and customs union, finance minister Philip Hammond and others have this month echoed calls by businesses for less of a "hard Brexit" and retaining closer customs ties.
With discontent in europhile Scotland and troubled Northern Ireland, which faces a new EU border across the divided island, Brexit poses new threats to the integrity of the United Kingdom.
It will test the ingenuity of thousands of public servants racing against the clock to untangle 44 years of EU membership before Britain is out, 649 days from now, on March 30, 2019. For the officials sitting down on Monday, at least on the EU side, a major worry is Britain crashing out into a limbo, with no deal.
For that reason, Brussels wants as a priority to guarantee rights for 3 million EU citizens in Britain and be paid tens of billions of euros it says London will owe on its departure.
With a further million British expatriates in the EU, May too wants a deal on citizens' rights, though the two sides are some way apart. Agreeing to pay a "Brexit bill" may be more inflammatory.
Brussels is also resisting British demands for immediate talks on a future free trade arrangement. The EU insists that should wait until an outline agreement on divorce terms, ideally by the end of this year. In any case, EU officials say, London no longer seems sure of what trade arrangements it will ask for.
But Union leaders, including German Chancellor Angela Merkel and French President Emmanuel Macron, are also determined not to make concessions to Britain that might encourage others to follow.
When 52 percent of British voters opted for Brexit, some feared for the survival of a Union battered by the euro crisis and divided in its response to chaotic immigration. The election of the fervently europhile Macron, and his party's sweep of the French parliament on Sunday, has revived optimism in Brussels. (Editing by Janet Lawrence)