In a verdict upholding a High Court decision the UK’s most senior court dismissed the Government’s appeal by a majority of eight to three.
While Jeremy Corbyn has repeatedly said his MPs will not seek to stop Brexit, the ruling does open up the possibility for Labour to table amendments to the Article 50 bill.
Speaking in the immediate aftermath of the Supreme Court’s verdict, Keir Starmer told Sky News: “We have said repeatedly we accept and respect the outcome of the referendum.”
“We have said all along the terms of the negotiation and the role of parliament is really important and we will be pushing for amendments,” he added.
Much of Labour’s concern centres on claims Mrs May could seek to turn Britain into a “bargain basement” economy by lower taxes and regulations to entice business.
Shadow Brexit Secretary Keir Starmer reacts to the Supreme Court ruling
We will be pushing for amendments
While Labour insists MPs will not obstruct or frustrate the will of the people, some die-hard Remainers have said they will not vote for Article 50.
David Lammy, MP for Tottenham, said back in November: “I will be voting with my constituents, I will be voting in what I believe to be the interests of the country – which I think all MPs will take onboard – and will absolutely not be voting to trigger Article 50.”
And Labour peer Peter Hain said earlier this week he will also vote against Brexit because of Theresa May’s “right wing” strategy.
Speaking on BBC News, he said: “I recognise the referendum result and respect it.
“But what we were not promised, as the prime minister has now done, is a Brexit deal that is extremely right wing, damaging to Welsh jobs and damaging to our most vulnerable people.
“I can’t in all conscience and principle vote for that.”
Ukip leader Paul Nuttall has warned MPs and peers not to hamper the passage of the legislation.
"The will of the people will be heard, and woe betide those politicians or parties that attempt to block, delay, or in any other way subvert that will," he said.
Ukip's sole MP, Douglas Carswell, said the ruling cleared the way for Mrs May to trigger Article 50.
Who is Gina Miller?
Thu, November 3, 2016
Gina Miller is leading the challenge that has blocked Theresa May triggering Article 50 without the consent of Parliament, take a look back at Miller's background.
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Gina Miller has won the case to block Theresa May from Article 50
"All the lawyers in London are not going to be able to stop us leaving the European Union," he told Sky News.
"We need two votes – one in the House of Commons, one in the House of Lords. If either of those chambers attempt to veto the will of the people, we can replace the Commons with a general election and we can certainly replace the Lords by creating 800 new peers."