Barnier said the backing of the remaining 27 countries was a further show of "determination and confidence".
Barnier said the backing of the remaining 27 countries was a further show of "determination and confidence" in what promise to be two years of bruising negotiations with London.
The former European commissioner and French foreign minister also warned his British counterpart David Davis against further threats to walk away from the talks if Brussels pushes a reported 100-billion-euro ($112-billion) divorce bill.
"We are ready and well prepared. We have a clear mandate supported by all 27 EU member states," Barnier told a news conference after EU ministers formally approved his tough negotiating mandate for the talks.
Barnier said the EU's new Brexit team would meet on Tuesday to finalise its negotiating position, which would be sent "very quickly" to London after the June 8 election in which Prime Minister Theresa May is seeking to strengthen her own mandate.
"I hope to organise the first round of negotiations as soon as possible, hopefully in the week of the 19th of June," Barnier said.
Maltese Deputy Prime Minister Louis Grech, whose island nation holds the rotating EU presidency, said Monday's decision "shows that the unity and consensus is being clearly continued" in the bloc.
May, who took over from David Cameron after last June's seismic Brexit vote and triggered the divorce process on March 29, gave a similar timeline for the start of talks.
"There are just 17 days to go until this crucial general election. Just 11 days after that, the European Union wants the Brexit negotiations to begin," she told a campaign event in Wales.
"There will be no time to waste and no time for a new government to find its way."
She added: "If we don't get this right, the consequences for the United Kingdom and for the economic security of ordinary working people will be dire. If we do, then the opportunities ahead are great."
Barnier's Brexit mandate comes from "negotiating directives" approved by the EU 27 ministers.
They are a toughened version of guidelines that EU leaders adopted in just four minutes -- one for each decade of Britain's membership -- at a summit on April 29 in a rare show of unity for the often divided bloc.
The EU insists on making "sufficient progress" on three key divorce issues before talks can start on a future UK-EU trade deal. These are the rights of EU 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.
Britain however wants the divorce settlement and the future relationship to be discussed in parallel.
The most contentious issue in the talks threatens to be the exit bill.
Brexit minister Davis at the weekend reiterated an earlier threat to quit the talks if the EU does not moderate its demands.
"We don’t need to just look like we can walk away, we need to be able to walk away. Under the circumstances, if that was necessary, we would be in a position to do it," he told the Sunday Times newspaper.
Barnier, who previously dealt with Davis when they were ministers in the 1990s, said quitting was not an option for the EU side.
He warned that Britain could in doing so also harm prospects for a trade deal, saying: "No one should forget the perspective of the new partnership."
EU ministers warned that talks would be tough as they stood firm on Britain's bill.
German foreign ministry state secretary Michael Roth said as he arrived for the meeting: "We all have to prepare for very difficult negotiations... the clock is ticking."
Luxembourg Foreign Minister Jean Asselborn warned Britain that while it will leave the EU in 2019, it could be paying up for years more on financial obligations it agreed to while a member of the EU.
"That goes until 2020 and of course beyond," Asselborn said, referring to the seven-year budget Britain agreed to with the other EU states in 2014.