In a landmark Brexit speech, Mrs May also pledged that other EU nations would not be left out of pocket by Britain's decision to pull out, paving the way for an estimated payment of around Â£18 billion (about 20 billion euro) into Brussels budgets up to 2020.
Mrs May put no figure on the amount the UK will pay in its so-called "divorce bill" and stuck to her position that the final total cannot be agreed until the future trade relationship is settled. But she insisted that estimates of Britain's liabilities, which have ranged from Â£50-Â£80 billion, were "exaggerated and unhelpful".
Speaking in a Renaissance church in the Italian city of Florence, Mrs May rejected the models for the future UK-EU relationship offered by Norway's membership of the European Economic Area or Canada's free trade deal on goods, which came into effect on Thursday.
She called on fellow European leaders to show "creativity and flexibility" in forging a unique partnership, which would include a "comprehensive and ambitious" trade deal and a new treaty guaranteeing future co-operation on security, law enforcement and criminal justice.
"As we meet here today, in this city of creativity and rebirth, let us open our minds to the possible," said the Prime Minister. "To a new era of co-operation and partnership between the United Kingdom and the European Union. And to a stronger, fairer, more prosperous future for us all.
"For that is the prize if we get this negotiation right. A sovereign United Kingdom and a confident European Union, both free to chart their own course. A new partnership of values and interests. A new alliance that can stand strongly together in the world."
Mrs May's 35-minute address was hailed as "positive, optimistic and dynamic" by Foreign Secretary Boris Johnson, who was present in the audience just six days after publishing a 4,000-word personal Brexit manifesto which exposed Cabinet rifts over the future relationship with Europe.
Fellow Leave campaigner Michael Gove said the PM's "excellent" speech was delivering on the will of the British people.
Mr Johnson said the Prime Minister had "rightly" disposed of the Norway option, which would have seen Britain continuing to make payments into EU budgets and accept free movement of people in return for access to the single market and customs union.
But he accepted that the UK would have to wait to take back powers from Brussels, telling reporters: "As the Prime Minister rightly said we are going to have a transition period and after that of course we are going to be taking back control of our borders, of our laws, of our destiny."
The leader of the centre-right EPP grouping in the European Parliament Manfred Weber, a German MEP and close ally of Chancellor Angela Merkel, said: "In substance, PM May is bringing no more clarity to London's positions. I am even more concerned now."
And former Ukip leader Nigel Farage said Mrs May's vision meant the UK will "leave the EU in name only".
Restating her message in her Lancaster House speech in January, Mrs May said Britain would have to leave both the single market and customs union.
She did not repeat the mantra that "no deal is better than a bad deal" in her speech, but asked later whether this remained the Government's position, she told reporters: "We continue to believe that."
In a warning to EU leaders who must decide next month whether sufficient progress has been made in divorce talks to move on to discussion of the future relationship, Mrs May said that if no agreement was reached "it would be a failure in the eyes of history and a damaging blow to the future of our continent".
Accepting that neither the Government nor the EU will be ready to fully implement Brexit on March 29 2019, the PM proposed an implementation period during which "the existing structure of EU rules and regulations" would apply.
"As of today, these considerations point to an implementation period of around two years," she said, although in some areas changes to new arrangements could be made more quickly.
Mrs May said there would be a "clear double lock" to the implementation period, giving businesses the certainty to plan for change and a guarantee that the temporary transitional arrangements "will not go on forever".
In an attempt to break the deadlock over the UK's financial settlement, Mrs May promised the UK would honour its commitments under the existing budget period, which lasts until 2020, and continue to participate in some other programmes on areas including science, education and culture beyond Brexit.
"I do not want our partners to fear that they will need to pay more or receive less over the remainder of the current budget plan as a result of our decision to leave," she said.
"The UK will honour commitments we have made during the period of our membership. And as we move forwards, we will also want to continue working together in ways that promote the long-term economic development of our continent."
Mrs May said that her proposed treaty on security would be "unprecedented in its breadth, taking in co-operation on diplomacy, defence and security, and development. And it will be unprecedented in its depth, in terms of the degree of engagement that we would aim to deliver".