A petition calling for the Government to revoke Article 50 has had the highest volume of sign-ups on record, according to Parliament's official Petitions Committee.
More than one million people had signed the petition by early afternoon on Thursday, following increased activity on the website after the Prime Minister's speech on Wednesday night.
Revoke Article 50 began trending on Twitter following the speech and continued to be a global trend into Thursday.
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In her Downing Street statement, Mrs May blamed MPs for failing to implement the result of the 2016 EU referendum and told frustrated voters: "I am on your side."
The Petitions Committee said nearly 2,000 signatures were being completed every minute over Thursday lunchtime, crashing the website because of the unprecedented number of visitors.
The petition is the third most popular to be submitted to the Parliament website, surpassing the 100,000-signature threshold needed for it to be debated in Parliament.
A 2016 petition calling for a second EU referendum should the winning vote and turnout not reach a certain threshold has received the most signatories to dater at almost 4.2 million.
A petition to prevent Donald Trump from making a state visit to the UK is in second place with 1.9 million sign-ups.
By contrast, the most popular pro-Brexit petition on the Parliament website, which calls on the Government to "leave the EU without a deal in March 2019", had received almost 375,000 signatories by Thursday afternoon.
The website requires signatories to tick a box confirming they are a British citizen or UK resident and provide a name, email address, country and postcode.
Data from the petitions website shows 960,000 signatories claim to be from the UK, followed by France (8,300), Spain (4,600) and Germany (3,700).
The constituencies with the highest number of sign-ups are Bristol West (Labour, 6,900), Hornsey and Wood Green in London (Labour, 5,937) and Brighton Pavillion (Green, 5,800).
Signatures are not counted if people do not click a link in a confirmation email but they are not required to provide any proof of their address or citizenship.
Celebrities and MPs have tweeted their support for Parliament to revoke the Treaty of Lisbon clause that deals with leaving the EU.
Famous figures including actors Hugh Grant and Jennifer Saunders, TV presenter and author Caitlin Moran, physicist Brian Cox and former Labour press chief Alastair Campbell all urged their followers on social media to sign the petition.
Alongside the link, Grant wrote: "I've signed. And it looks like every sane person in the country is signing too. National emergency."
A House of Commons spokesman said: "We know that the petitions website has been experiencing problems due to the number of people using the site.
"This is a mixture of people signing petitions and refreshing the site to see changes to the number of signatures.
"The majority of people are now able to use the website and we and the Government Digital Service are working to fix any outstanding problems as soon as possible."