Members of the United Kingdom's House of Lords went head-to-head with Prime Minister Theresa May's government on Monday evening when they voted to give Parliament a bigger role in Brexit.

Their amendment to the EU Withdrawal Bill would let MPs stop the UK from leaving the European Union if no acceptable agreement is reached.

Liam Fox, Britain's international trade secretary, told the BBC the unelected Lords' decision, reached by 335 votes to 244, could see the UK stay in the EU "indefinitely".

He said members of Britain's second chamber were trying to "thwart the will of the British people" by finding a back door way to block Britain's pending withdrawal following the June 2016 referendum.

"The public voted in the referendum," he told BBC Radio 4's Today program. "They voted to leave the European Union, the House of Commons passed the legislation to allow that to happen. It is not acceptable for an unelected house to try to block the democratic will of the British people."

But the Labour Party said the vote marked a "hugely significant moment" in efforts to ensure Parliament has a "proper role" in Brexit negotiations.

Following the House of Lords vote, MPs could send May and her negotiators back to Brussels to look for a deal everyone can live with.

But, despite the Lords' stance, MPs do not have to accept the amendments. The amendments will now return to the Houses of Parliament where MPs could defeat them.

The Independent newspaper said on Tuesday that, if MPs do not defeat the amendment, the door will be opened to the possibility of a fresh Brexit referendum.

The UK plans to leave the EU on March 29, 2019 and enter a transitional phase until Dec 31, 2020, after which the new permanent relationship will follow.