Theresa May has insisted Brexit WILL happen - despite being delayed yet again until Halloween.
It's the second time Britain's departure from the EU has been delayed in just over a month.
Confirming the delay, European Council President Donald Tusk urged the UK: "Do not waste this time."
And he suggested the UK could still decide to cancel Brexit altogether.
Mrs May told a press conference following the announcement that it was still her hope that a deal would be agreed early enough that Britain would not have to take part in EU elections next month.
She said: "If we are able to pass a deal in the first three weeks of May, we will not have to take part in European Elections and will officially leave the EU On Saturday June 1."
Tonight's agreement states that if the UK does not take part in the elections, it will be forced out of the EU without a deal that day.
Asked why the British people - and her own party - should believe Brexit will ever happen, Mrs May said: "I'm very clear that we must deliver as a government on Brexit.
"We gave the choice to the people in the referendum. We said we would abide by the decision that the British people took.
"It is now our duty to abide by that decision."
She went on: "We could already have left the European Union.
"What I want to do now is to work to ensure that we get that majority in parliament, so that we can ratify this agreement, leave the European Union, do it in an orderly way, with a deal and do it as soon as possible."
Mr Tusk confirmed that the UK would remain a "full member" of the EU during the extension until October 31.
And he said the extension could be ended early by Mrs May getting a deal approved - or by cancelling Brexit altogether.
In an earlier Press Conference, Mr Tusk said the extension "means an additional six months for the UK.
"During this time the course of action will be entirely in the UK's hands."
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He said the UK could use the time to get Theresa May's deal through, and can terminate the extension early if one is agreed.
He added: "It can also reconsider the whole Brexit strategy that can lead to changes in the political declaration - though not in the Withdrawal agreement."
And he said Britain could still decide during the six months to withdraw Article 50 and cancel Brexit altogether.
"This extension is as flexible as I expected - and a little bit shorter than I expected," Mr Tusk said, in a message to his "friends in the UK."
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"But it is still long enough to find the best possible solution.
"Please do not waste this time."
European Commission President Jean-Claude Juncker said there would "probably" be European Elections in the UK.
"That may seem a bit odd," he said. "But rules are rules."
He said: "No deal would be a disaster and everybody knows that", adding everyone would be working to avoid such an outcome.
