In a historic, detailed speech, Mrs May dashed the hopes of Remainers and delighted Brexiteers by setting out a vision of an independent UK – a "trading nation" that will look beyond Europe to "new friends and old allies".

She confirmed the UK would leave the European single market and instead would negotiate a new free trade deal with the European Union.

She was less clear on the UK's role in Europe's customs union, saying Britain would not be part of the common commercial policy or external tariffs, but could remain a "signatory to elements" in order to lower barriers to trade.

She emphasised the British people had voted to regain control of immigration, though she wanted European citizens already in the UK to retain their right to work there.

She revealed that the final Brexit deal would be presented to the British Parliament for a vote before it comes into force – though she did not discuss what would happen if it was voted down, saying instead she was confident it would be approved.

Mrs May concluded with a barely veiled threat to Europe: that if they sought a "punitive" Brexit then it would be "an act of calamitous self-harm" as the UK would retaliate by lowering its tax rates below Europe's to draw companies and investors from the continent.

She also admonished Europe for "trying to hold things together by force, tightening a vice-like grip that ends up crushing into tiny pieces the very things you want to protect".

But she said it was "overwhelmingly and compellingly in Britain's national interest that the EU should succeed".

Mrs May's speech drew a rapturous reception from those who had campaigned for Brexit, with UKIP leader Nigel Farage commenting, "I can hardly believe that the PM is now using the phrases and words that I've been mocked for using for years. Real progress."

But it is likely to be less welcome in Europe.

European ambassadors were present in the room for the speech, and Fairfax Media understands few if any applauded the conclusion.

Some were seen shaking their heads when it came to the threat of Britain becoming a tax haven.

In a deliberate irony the speech took place at Lancaster House in London, where Margaret Thatcher set out her plan for the UK's membership of the single market in April 1988.

Mrs May began by saying she wanted the country to be "a truly global Britain", a "best friend and neighbour" to Europe but one that "goes out into the world to build relationships with old friends and new allies alike ... a great global trading nation respected around the world".

She said the UK would begin by adopting EU laws, then modifying them, to allow a smooth transition and certainty for business.

Mrs May also spent some time reassuring Ireland that there would be a "practical solution" that balanced the common travel area with Northern Ireland with the need to protect the UK's new immigration controls.

"Nobody wants to return to the borders of the past," she said.

On immigration, she said the UK would "get control" of the number of people coming to Britain from the EU, but would guarantee the rights of EU citizens already in Britain as an "important priority", preferably before the rest of the Brexit deal was done.

On trade, she said "as a priority, we will pursue a bold and ambitious free trade agreement with the European Union.

"This agreement should allow for the freest possible trade in goods and services between Britain and the EU's member states. It should give British companies the maximum freedom to trade with and operate within European markets – and let European businesses do the same in Britain.

"But I want to be clear. What I am proposing cannot mean membership of the single market."

This agreement might take in elements of the current single market as it "makes no sense to start again from scratch", she said.

But the UK needs to leave the single market and key elements of the customs union, Mrs May said, in order to strike its own trade deals.

"We want to get out into the wider world, to trade and do business all around the globe."

"Countries including China, Brazil, and the Gulf states have already expressed their interest in striking trade deals with us. We have started discussions on future trade ties with countries like Australia, New Zealand and India. And President-elect Trump has said Britain is not 'at the back of the queue' for a trade deal with the United States, the world's biggest economy, but front of the line."

On timing, she said she wanted a full agreement with the EU by the end of the two-year Brexit process, which is expected to begin in March.

But she conceded it would be followed by a "phased process of implementation", where some of the new arrangements would be introduced quickly and others "might take longer".

Australia's high commissioner to the UK Alexander Downer said it had been a "historic speech", which had made clear what had been implicit in the government's previous announcements on Brexit.

"We certainly welcome the fact that it is possible to negotiate a free trade agreement with the UK that will be in our best interests," he said. "Now she has said Britain will not remain in the single market, we know we can continue to do scoping work on a free trade agreement with the UK, and we look forward to the negotiations after March 2019."