By this morning, as many as 3,048,000 people had signed the petition on the official UK Parliament website. The figure takes it well over the 100,000-signature threshold required to trigger a debate in the House of Commons.
Ben Howlett, a Conservative MP, confirmed on Twitter that the petition will be discussed on Tuesday by the House of Commons petitions Select Committee.
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On Twitter, he wrote: "We can stop this madness and bring this nightmare to an end through a vote in Parliament. Our sovereign Parliament needs to now vote on whether we should exit the EU."
"The referendum was was an advisory, non-binding referendum. The Leave campaign's platform has already unravelled and some people wish they hadn't voted to Leave," Lammy said.
"Parliament now needs to decide whether we should go forward with Brexit, and there should be a vote in Parliament next week. Let us not destroy our economy on the basis of lies and the hubris of Boris Johnson," he added.
The development came as some voters who had backed the "Leave" campaign took to Twitter to register their regret - adopting the #Regrexit hashtag.
A "Leave" voter told the London Evening Standard she would change her vote if given the opportunity.
"This morning the reality is actually hitting in and the regret is hitting in," she said. "I wish I had the opportunity to vote again, simply because I would do things differently."
The petition, started by William Oliver Healey, states: "We the undersigned call upon (the UK) Government to implement a rule that if the Remain or Leave vote is less than 60%, based on a turnout less than 75%, there should be another referendum."
Demand for a fresh vote was so high that it crashed the parliament.Uk website on Friday as the petition was shared widely on social media and new signatures were added at a rate of more than 1,00,000 an hour.
Meanwhile, a second petition urging London mayor Sadiq Khan to declare the British capital's independence has also crossed 100,000 signatures.
However, this petition was on the 'Change.Org' platform rather than the official Parliament.Uk website.
Set up by James O'Malley, it read: "Let's face it - the rest of the country disagrees. So rather than passive aggressively vote against each other at every election, let's make the divorce official and move in with our friends on the continent."
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