The bloc's unelected bureaucrats are still intent on forcing the United Kingdom into agreeing to a "backstop" if the border can not be kept completely open, which would cut the province off from the rest of the United Kingdom by locking it in to the bloc's Single Market.
"No technology solution to address these issues has been designed yet or implemented anywhere in the world, let alone in such a unique and highly sensitive context as the Northern Ireland border", she said.
"Every country will struggle to put in place the necessary infrastructure and customs and veterinary officials in their ports and airports".
The prime minister has begun a two-day visit to Northern Ireland in Belleek, County Fermanagh.
"The reason we say that is because overflight rights are not guaranteed by the European Union, rather by multilateral treaty which both ourselves and Ireland have signed up to".
Her speech comes at the end of another tense week for May after months of Brexit crisis that included ministerial resignations and close votes, and is ahead of the summer recess next week.
The sticking point in the EU-UK negotiations, so far, has been around the proposed customs backstop - an arrangement that will apply to the Irish border after Brexit, if a wider deal or technological solution can not keep it as frictionless as it is now. "But to evolve their position in kind", told the audience at Belfast's Waterfront Hall.
May says her proposal - to treat trade in goods and services differently - would keep "frictionless" trade and make a border between the Republic of Ireland - an European Union member - and Northern Ireland unnecessary.
"And as they made clear this week, it is not something the House of Commons will accept either".
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Mrs May will promote the Chequers plan, and a subsequent government white paper, as the way to the stronger and brighter tomorrow.
"And, on that basis, I look forward to resuming constructive discussions".
"We've only got 12 weeks really left to nail down the details of the agreement", he added.
The UK has a "duty" to ensure that its borders with neighbouring countries function smoothly, she said, adding that this was "a particular challenge" in Northern Ireland.
At the Irish border, business leaders have expressed concern that any division, even as a result of a soft Brexit, would have consequences.
If "no deal" produces a hard border, it will not be better than a "bad deal" for people living here.
It is unfortunate that people are trying to undermine the Brexit white paper.
According to The Times, European governments are firmly against Mrs May's complicated customs plan but divided on allowing the United Kingdom access to the Single Market in return for following EU rules on goods, as her white paper proposes.
Ms Leadsom added: It needs to be quite clear to the European Union, "you've got to come to the table and start negotiating with serious focus and concentration and goodwill because otherwise, we will be leaving with no deal".