The Taoiseach has said the onus is now on Westminster to come up with solutions that they can support and that the European Union and Ireland can accept.
Leo Varadkar said the Irish Government profoundly regrets the decision of the House of Commons to reject the Brexit deal last night.
Mr Varadkar said Irish plans for a no-deal Brexit are continuing and are no longer contingency, but being implemented by the Government.
He added that businesses and other organisations must start doing the same.
However, Mr Varadkar said that that a no-deal scenario could still be avoided, including, if neccessary, an extension of the Article 50 deadline.
Taoiseach: "The onus is on Westminster to come up with solutions that they can support." | pic.twitter.com/dc7kQltkS6— RTÉ News (@rtenews) January 16, 2019
Earlier the Tánaiste and Minister for Foreign Affairs said it was still unclear as to what the British parliament is looking for, following last night's rejection of the Brexit deal.
Simon Coveney said Ireland would not object if Britain asked for an extension to Article 50, but that such a request would have to be accompanied by a plan that allowed for an orderly Brexit.
Speaking on RTÉ's Morning Ireland, Mr Coveney said that a withdrawal deal was still on the table and he did not believe the EU was in any mood to change it.
The Tánaiste reiterated that Ireland must hold its nerve in the weeks ahead.
Mr Coveney said the British Prime Minister and the Irish Government had committed to ensuring there was no physical border on the island of Ireland and that no one had put forward a viable alternative solution to the backstop.
He also said he did not believe there would be any food or medicine shortages in the event of a no deal Brexit and there was no need for people to "stockpile".
Mr Coveney said: "There is so much still to happen in Westminster in the next few days before it becomes clear what Britain is asking for.
"It's hard for us as EU leaders to respond in a helpful way if we still don't know what they are asking for.
"We need to hold our nerve this week. There's a motion of no confidence today, I believe that will be lost, and my understanding is that the Prime Minister's intention then is to try and reach across parties to try and find a way of getting a majority in Westminster to support a new approach.
"When we understand what that new approach is, which we get next Monday, we'll try and respond in a helpful way."
Mr Coveney noted that the Withdrawal Agreement was signed by 28 governments including Britain.
"I don't think the EU is in any mood to change the Withdrawal Agreement significantly at all," he added.
"We don't even know what the ask is, we'll have to wait and see what they're asking for.
"Those who have rejected the backstop do not have alternative solutions in terms of how we provide reassurance (on the border)."
He added: "If there isn't a future relationship that resolves the border issue, then the backstop will kick in which involves minimal checks on goods moving from Great Britain into Northern Ireland.
"The position is we support the backstop. The Irish Government have made very clear that we are not going to start talking about alternatives to the backstop. It took us two years to negotiate an approach that can provide the guarantees that we have given to people.
"We're not going to allow physical border infrastructure to reappear."
Sinn Fein's finance spokesperson Pearse Doherty called on the Government and the EU to stay firm in relation to the backstop, adding "what we need to be very, very clear about is that there is no good outcome to Brexit."
He said that the backstop is the "only way we can mitigate the damages of Brexit and there cannot be any diminution of the Irish backstop in relation to any talks or negotiations that will happen.
"If the British want to leave the EU, then so be it. But they need to ensure that the Irish people, the Irish economy and the Good Friday Agreement is protected, and the best way of doing that is making sure the backstop is part of any withdrawal agreement."
Meanwhile, the DUP's Brexit spokesperson he did not fear a no deal Brexit and did not think Theresa May had room for manoeuvre on extending Article 50 because she would face an even bigger revolt.
He also defended comments made yesterday by his party leader Arlene Foster that there had never been a hard border on the island of Ireland.
He said the prime minister had made it clear that the UK was leaving the EU on 29 March and she was not bringing forward legislation to change that date.
Former Northern Ireland secretary Theresa Villiers said Mrs May had to engage with EU leaders and to explain that the Westminster parliament had shown very clearly that the Brexit deal was not acceptable and that it is acceptably dead in its current form and needed to be significantly changed.
Speaking on the same programme, she said she did not believe that the backstop to prevent a hard border in Ireland after Brexit was necessary.
Ms Villiers said: "It's clear to me that we can maintain a free-flowing, invisible border on the island of Ireland without the compromises of the backstop.
"I and my colleagues in the European Research Group put forward proposals back in September that demonstrate that you can use existing flexibilities in the EU customs code, existing administrative procedures of technology to ensure that new customs and compliance checks are done away from the border so they don't require new physical infrastructure at the border."
Ms Villiers said it was in the interests of both sides that there was an orderly exit, but it was hard to say yet whether that will be achieved before 29 March or afterwards.
Additional reporting Laura Hogan, PA