Video report by ITV News Political Editor Robert Peston
Theresa May has told EU leaders that both sides must be willing to compromise as she called for the "deepest and broadest possible" trade deal after Brexit.
The Prime Minister said it was time to face some "hard facts" as the UK prepares to split from the bloc.
"We're leaving the single market. Life is going to be different," she said. "We need to strike a new balance".
But she said Europe would also be harmed if they fail to reach an agreement securing close cooperation after Brexit, as she made a highly-anticipated keynote speech attempting lay the ground for the next phase of negotiations.
"We both need to face the facts that this is a negotiation and neither of us can have exactly what we want. But I am confident that we can reach agreement," she said.
Speaking at the Mansion House in the City of London, Mrs May set out "five tests" to guide Britain's approach to the continuing negotiations.
Reciprocal binding commitments to ensure fair and open competition
A completely independent arbitration mechanism to resolve disputes
An ongoing dialogue with the EU, with regular consultation
An arrangement for data protection
Measures to maintain links between people
Mrs May pushed for close cooperation in a host of sectors as she said Britons had voted for "a new and different relationship with Europe" but the UK's shared goals with the continent have not changed.
She again stressed there must be a bespoke deal, rejecting models based on either Norway or Canada as not good enough.
The UK must strike a "new balance" with the EU she said, as she acknowledged that it can’t enjoy “all the benefits with all of the obligations”, she said.
But she warned it won’t accept the “rights of Canada but the obligations of Norway".
She hit out at European accusations of "cherry-picking".
"The fact is that every free trade agreement has varying market access depending on the respective interests of the countries involved," she said.
"If this is cherry-picking, then every trade arrangement is cherry-picking."
ITV News Political Editor Robert Peston believes that once Brexiteers look in detail at what Mrs May has set out "they may become a little bit more uncomfortable"
Her speech comes after European Council president Donald Tusk warned that her insistence on taking Britain out of the single market and the customs union meant it could not enjoy "frictionless trade" with the EU after Brexit.
Mrs May insisted that a solution must be found that avoids a hard border with Northern Ireland.
"I am not going to let our departure from the European Union do anything to set back the historic progress we have made in Northern Ireland, nor will I allow anything that would damage the integrity of our precious union," she said.
She has earlier rejected suggestions from the EU that could see it diverge from London and align with Brussels' trade rules.
The Prime Minister rejected the suggestion from some Brexiteers that the UK could unilaterally decide not to impose a hard border.
She said: "It is not good enough to say 'we won't introduce a hard border; if the EU forces Ireland to do it, that's down to them'.
"We chose to leave; we have a responsibility to help find a solution.
"But we can't do it on our own. It is for all of us to work together."
ITV News Political Correspondent Carl Dinnen explains how Theresa May attempted to appease both Leavers and Remainers with her speech
Mrs May acknowledged there would be "ups and downs over the months ahead" but said she was "confident" an agreement could be reached.
"The world is watching," she said.
"We should not think of our leaving the EU as marking an ending, as much as a new beginning for the United Kingdom and our relationship with our European allies.
"Change is not to be feared, so long as we face it with a clear-sighted determination to act for the common good.
"Nor is Brexit an end in itself. Rather, it must be the means by which we reaffirm Britain's place in the world and renew the ties that bind us here at home.
"And I know that the United Kingdom I treasure can emerge from this process a stronger, more cohesive nation.
"This is an optimistic and confident future which can unite us all."
Mrs May's speech was praised by prominent Brexiteer Boris Johnson, and her insistence there could no be division of the UK was welcomed by DUP deputy leader Nigel Dodds.
However, she was lambasted by Labour, with Angela Rayner saying it was "like your careers adviser sitting you down after you have been told your projected grades aren't good but trying to make the most of it".