At a Brussels press conference, Mr Barnier told reporters the agreement was "a demonstration of determination and confidence" backed not only by all 27 national governments, but also by the European Commission, European Council and European Parliament.
In addition to naming Barnier, a veteran French public official, as negotiator, ministers approved the formation of a Brexit task force within the Council of the European Union, to be led by Didier Seeuws, a Belgian advisor to Council President Donald Tusk.
The meeting in Brussels will see the Europe ministers of the EU's remaining 27 countries review and possibly revise the mandate for their appointed negotiator, Michel Barnier.
The talks will quickly center on the tens of billions in costs London would be expected to pay for the divorce.
The first round of talks will start on 19 June and Mr Barnier will report to European Union leaders at a summit three days later.
The relocation away from London of the European Banking Authority and European Medicines Agency.
British Prime Minister Theresa May, who triggered the divorce process on March 29, said "the next five years will be among the most challenging in our lifetime".
Monday's explicit language on the payments Britain should make even included "a schedule of payments" from London.
Barnier said he would propose to his British counterpart that the discussions take place in four-weekly cycles. The deal we seek will be negotiated by me or Jeremy Corbyn. "Moreover, there is also a very limited timeframe which adds pressure to an already hard process". "We have two years time, the clock is ticking and we have to start working focused".
"I hope to organise the first round of negotiations as soon as possible, hopefully in the week of the 19th of June", Barnier said. "[No deal] is not an option", said Barnier, "my option is to succeed". "Which is risky, because once you walk away, you need a major concession to come back to the table and we are simply not able to provide any".
Barnier sought to reassure the United Kingdom that his goal was to move to discussion of the future relationship as quickly and effectively as possible. "It is their choice and responsibility".
These are the rights of European Union citizens in Britain and British citizens on the continent; London's exit bill; and arrangements for the border between Ireland and the British province of Northern Ireland.
The EU's negotiator wants to reach agreement on the key issues of the first phase by the end of this year. "There will be no time to waste and no time for a new government to find its way", she said in a speech in Wales.
May's Conservative Party stated in their election manifesto that they would be willing to walk away without a deal if an agreement meant a bad deal for Britain.
Ten tips that help to win
Tip # 2 You need to find out as much information as you can on the online casino before you will give away your money. Unlikely you will want to swim in the dark pond if you have no idea what is waiting for you below the surface.
DHS hopeful Clarke denies plagiarism in master's thesis
His thesis, "Making U.S. security and privacy rights compatible, failed to attribute his sources properly a minimum of 47 times". Clarke told the Milwaukee Journal-Sentinel on Sunday that "only someone with a political agenda would say this is plagiarism".