Warning the EU that she would continue preparations for a “no-deal” Brexit, the prime minister said the UK “stands ready”.
In a statement in Downing Street, she said: “I have treated the EU with nothing but respect, the UK expects the same”.
She added: “The EU should be clear: I will not overturn the result of the referendum, nor will I break up my country.
“We cannot accept anything that does respect the referendum.
“Anything which fails to respect the referendum or which effectively divides our country in two would be a bad deal and I have always said no deal is better than a bad deal.
“But I have also been clear that the best outcome is for the UK to leave with a deal. That is why, following months of intensive work and detailed discussions, we proposed a third option for our future economic relationship, based on the frictionless trade in goods.
“That is the best way to protect jobs here and in the EU and to avoid a hard border between Ireland and Northern Ireland, while respecting the referendum result and the integrity of the United Kingdom.”
“Yesterday Donald Tusk said our proposals would undermine the single market. He didn’t explain how in any detail or make any counter-proposal. So we are at an impasse.”
The prime minister was dealt a large setback on Thursday after she failed to attract EU backing for her Brexit plan during an informal gathering of the bloc’s leaders in Salzburg, Austria.
At the end of the summit, European Council president Mr Tusk admitted there were “positive elements” to Mrs May’s Chequers proposals but poured cold water over the prime minister’s outline for the future EU-UK trade relationship.
Rejecting Mrs May’s plan, he said: “The suggested framework for economic co-operation will not work.”
In her statement, Mrs May said: “At this late stage in the negotiations, it is not acceptable to simply reject the other side’s proposals without a detailed explanation and counter proposals.
“So we now need to hear from the EU what the real issues are and what their alternative is so that we can discuss them. Until we do, we cannot make progress.
Under the Chequers plan, as agreed by ministers in July, the prime minister hopes to agree a “common rulebook” between Britain and the EU on goods, as well as a new customs arrangement with the bloc.
But the EU is hostile to both elements, with Mr Tusk adding the plan “risks undermining the single market”.
Mrs May reassured EU citizens living in the UK that whatever the outcome of Brexit, their rights to live and work here would not be affected.