But Mrs May also had a message for Remainers, stressing how one of the five key Brexit tests she had set herself was to deliver on the public's historic decision of 2016, which meant the United Kingdom would be leaving the single market and customs union and that, while there might be regulatory convergence to begin with, Westminster would, after Brexit Day, be the sovereign power and could change course in future.
"We are close to agreement on the terms of the implementation period which was a key element in the December deal". We understand your principles.
But she emphasised that both sides stand to gain from the successful negotiation of a new partnership, whereas the European Union still sees Brexit not as a win-win but as a lose-lose.
She insisted she was "confident" a deal was "achievable", with Mrs May telling Brussels: "Let's get on with it".
She vowed that she would not walk away from talks over the coming months.
May said she stood by commitments she has made previously on the border issue.
On the Irish border specifically, she said: "Just as it would be unacceptable to go back to a hard border between Northern Ireland and Ireland, it would also be unacceptable to break up the UK's own common market by creating a customs and regulatory border down the Irish Sea".
"What would be cherry-picking would be if we were to seek a deal where our rights and obligations were not held in balance".
The prime minister, weakened at home and under pressure overseas, said that Britain sought the "broadest and deepest possible agreement, covering more sectors and cooperating more fully than any free trade agreement anywhere in the world today".
She said both sides could continue to recognise each other's trusted "traders schemes" and use technology to monitor larger vehicles at the border. "We must not only negotiate our exit from an organization that touches so many important parts of our national life, we must also build a new and lasting relationship while, given the uncertainty inherent in this negotiation, preparing for every scenario".
She said she would be guided by five tests including respecting the result of the 2016 Brexit referendum and reaching a solution that can endure.
Mrs May said: "The world is watching".
"We are leaving the single market", she said.
"Change is not to be feared, so long as we face it with a clear-sighted determination to act for the common good. This is an optimistic and confident future which can unite us all", she declared.
The head of a hardline group of pro-Brexit lawmakers on Saturday gave a cautious welcome to Prime Minister Theresa May's plans for Britain's future ties with Europe, saying now was not the time to nitpick.
"As in any negotiation, no-one will get everything they want".
Mrs May said Britain would not be buffeted by European Union demands in divorce talks, nor would it threaten to walk out of the negotiations.
"We have to accept that both sides are going to have to make compromises", he said.
"We will move forward by calm, patient discussion of each other's positions". "The prime minister's proposals are non starters", he said.
"By following the course I have set out today, I am confident we will get there and deliver the right outcome for Britain and the European Union".
She promised "broad" co-operation on energy, including a "close association" with Euratom, as well as continuity of transport services.