Britain was not invited to the special summit of 27 EU nations, where the leaders finalized the guidelines for two years of what are set to be grueling divorce talks.
European Council President Donald Tusk praised the "outstanding unity" of the 27 nations at a news conference concluding the summit, saying it had given the EU a "strong mandate" and the guidelines had been adopted unanimously and immediately.
In what he described as "unity in action," European Commission President Jean-Claude Juncker tweeted it had taken less than 15 minutes to approve the principles.
French President François Hollande said Brexit would be negotiated to meet Europe's interests -- and that there would be a cost for Britain.
"Obviously, there will be a price and a cost to be paid by the United Kingdom," he said. "This is their choice. It mustn't be punitive, but at the same time it is absolutely clear that Europe will have to defend its interests and that the UK -- outside of Europe -- will have a less good position than inside as it does today."
German Chancellor Angela Merkel -- who warned Britain this week not to hold any "illusions"
about what Brexit means for its future -- told reporters Saturday there was "no conspiracy" against the UK. "British citizens have held a referendum, and we have to accept that," she said, adding the exit was regretted by all 27 states.
The UK government had hoped to be able to negotiate a new trade deal with the EU at the same time as carrying out the complex process of unraveling a relationship lasting more than 40 years.
But the EU has insisted that progress must be made on key issues around Britain's continued budgetary commitment to the bloc, the future status of EU citizens living in Britain and the border between Ireland and Northern Ireland before talks on future relations with Britain can begin.
Juncker reiterated that position Saturday and stressed it would be for the EU leaders unanimously to decide when enough progress had been made on the "divorce talks" to proceed to trade negotiations.
He acknowledged it would be hard for the bloc to preserve the unity shown in agreeing to the guidelines throughout the negotiations, saying there would be "tough decisions" ahead, particularly in discussions of budgetary matters.
EU says it will act as one
A European Council statement on the adoption of the guidelines
warned of the potential disruption Brexit will cause to UK and other EU citizens' lives and businesses and said for this reason it will pursue a "phased" approach that prioritizes an "orderly withdrawal."
The EU will "maintain its unity and act as one" throughout the negotiations, with the aim of reaching an agreement fair for all its citizens and member states, it said.
"The Union will work hard to achieve that outcome, but it will prepare itself to be able to handle the situation also if the negotiations were to fail," the statement said.
An EU source earlier told CNN that there was increasing unity among the EU states. "The 27 want a deal and they realize the more united they are, the higher chances to get a deal," the source said. "We are not aligning against anyone. We are aligning (in) the interest of the 27, and then we need to find common ground with the British."
Bill due for Brexit
Last month, Juncker told the BBC that Britain would need to come up with roughly 50 billion pounds ($62.4 billion) as it leaves the EU to honor its financial obligations.
But the UK government has indicated it does not expect to pay nearly that much. Brexit Secretary David Davis told the BBC last month that Britain would meet its international commitments but that the bill would be "nothing like" the tens of billions of euros suggested by Juncker and others.
EU member states pay into a communal budget, which finances infrastructure projects, social programs, scientific research and pensions for EU bureaucrats. The budget is negotiated to cover a period of years, with the current agreement extending to 2020.
Britain pays roughly 10 billion pounds a year ($12.5 billion) more into the budget than it receives in benefits -- a fact often cited by supporters of Brexit -- and its departure will leave a large hole in the EU's finances.
However, EU foreign policy chief Federica Mogherini said Britain's departure would not "affect significantly" the EU's global position as a leader in trade, humanitarian affairs and security.
Northern Ireland status
The EU has also said that resolving the thorny issue of the Irish border
must be a priority in the upcoming negotiations.
In a letter to the 27 EU leaders Friday, Tusk said Europe should aim to avoid a "hard border" between the Republic of Ireland, which will remain in the EU after Brexit, and Northern Ireland, which leaves as part of the UK.
Border controls between the north and south were eased as part of the Good Friday Agreement, the 1998 accord that brought peace to Northern Ireland after decades of sectarian conflict.
Ireland asked Saturday that Northern Ireland be allowed to enter the EU automatically if the two Irelands ever unite, an EU source said. The request was approved but won't be published in EU minutes until the next European Council meeting in June, the source said.
The EU is not taking a stance on unification, which would be decided by the people of Ireland and Northern Ireland in accordance with the Good Friday Agreement.
A majority in Northern Ireland voted to remain in the EU in last year's UK referendum. At present, the "soft" border with the neighboring Republic of Ireland facilitates trade and the movement of people.
Both the UK and Irish governments want to maintain the soft border but have struggled to define how such an arrangement could work after the introduction of what would be the UK's only land border with the EU.
The issue has forced a discussion on the future status of Northern Ireland: The Good Friday Agreement stipulates that if polls show support for a referendum on unification, then the UK government must offer one. However, a poll by Ipsos MORI in Northern Ireland
in September did not suggest great enthusiasm for a united Ireland.