"A fair deal is possible and far better than no deal", Barnier said at a press conference on Monday.
The next set of talks will begin on July 17.
It's safe to say that the score is 1 for the European Union and 0 for the UK!
October 19 - Mr Barnier is set to report back to the EU27 on whether sufficient progress has been made to move on to phase two of the talks, covering the UK's future trade relationship with the remaining EU. He also predicted the start of UK-global trade talks by September 2016, and that German industry would pressure their government to get a special deal for Britain.
Merkel said she wanted the talks to be conducted "in a good spirit" and that Britain's position would become evident in the coming months.
Top negotiators Barnier and Davis have declared the talks have gotten off to a "promising start".
But Brussels has made clear that it expects all EU citizens in the United Kingdom to retain their full rights - and for these to be underpinned by the European court of justice.
Her announcement came at the end of a day which saw European Union leaders agree to pursue closer defence co-operation and threaten legislation to force internet companies to remove extremist material from websites.
May was hoping to increase her support in the British election on June 8, but instead her party lost its outright majority and now has to try to form a workable government with a tiny party from Northern Ireland.
"Today we agreed on dates, we agreed on organisation and we agreed on priorities for the negotiation", Barnier said.
These include the issues of Britain's estimated €100 billion (S$175 billion) exit bill, the rights of European Union citizens living in Britain, and the fate of the border in Northern Ireland.
But more than two months after Mrs May started the Brexit process by triggering Article 50 of the Lisbon Treaty and nearly a year after the European Union referendum result, M. Barnier warned "the clock is ticking" on the negotiations. And it was useful for me to sit down with my counterpart, David Davis. "We want to make sure that the withdrawal of the United Kingdom happens in an orderly manner", Barnier elaborated.
The two sides agreed to initially hold a week of negotiations every month until a new structure is defined.
The two parties also agreed on holding one-week monthly rounds of negotiations, noting that the next round will be held on July 17 and will tackle the main issues of Britain's separation from the European bloc.
"The other problem in parallel with that is that without creating a new hard border, at the same time we have to come up with solutions - especially for goods and services - but in a way that is compliant with the normal rules and integrity of the internal market", he said. He added: "Brexit won't make anything better, but it will make a lot of things more hard".
"The European council of 28 is not a place for negotiations with the United Kingdom".
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