Foreign Minister Simon Coveney today urged Theresa May to find a solution to the issue, which has threatened to derail Brexit talks.
He accused Mrs May’s government of failing to provide “credible and real answers” to Dublin’s concerns about the future of the border.
EU officials have raised fears that a physical border being set up post-Brexit could throw the Irish peace process into jeopardy.
Speaking at Fine Gael’s party conference in County Cavan, Mr Conveney said Dublin’s position was “consistent, firm and stubborn”.
He added: “Brexit the most important negotiation of our time.
“It is going to be what shapes our relationship with the European Union and the UK.
“We have an interwoven relationship with the UK which simply cannot be undone.
“We need a future trading relationship with Britain. Our relationship and our economic relationship will remain strong.
“As the Taoiseach [Leo Varadkar] said last night, on this island we are in the business of building bridges not borders.
“Britain does not have the right on its own to shape our future as Ireland in the context of the relationship that we have with the UK.”
Irish officials are believed to have demanded firm assurances over the border question before the UK can start trade talks with the EU.
According to a leaked European Commission document, Dublin wants a guarantee from Mrs May before a major EU summit in December.
The dossier also called for any divorce deal to protect “the integrity of the internal market and the customs union”.
A senior EU sources said the Irish position reflected the “guiding principles” of the EU’s approach to the problem.
Brexit Secretary David Davis yesterday said the UK was “committed” to opposing any possible hard border with Ireland.
He told reporters in Brussels: “Let me be clear, we cannot have anything resulting in a new border being set up with in the UK.
“We remain firmly committed to avoiding any physical infrastructure.
“We respect the EU desires, but they cannot come at the constitutional integrity of the United Kingdom.”