The move - which is expected to require emergency legislation - came after thousands of people were prevented from registering by the original deadline of midnight on Tuesday when a Government website crashed.
A statement was being made to MPs by Leader of the Commons Chris Grayling this afternoon to set out the procedure for implementing the change.
Cabinet Office minister Matt Hancock said: “Following discussions with the Electoral Commission and strong cross-party support expressed in the House of Commons, we will introduce secondary legislation to extend the deadline for voter registration until midnight tomorrow.
“Having taken the decision today, we think it is right to extend to midnight tomorrow to allow people who have not yet registered time to get the message that registration is still open and get themselves registered.”
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Brexit campaigners accused the Government of trying to “skew” the result of the EU referendum by extending the deadline.
The chief executive of the Vote Leave campaign, Matthew Elliott, said the Government was “trying to register as many likely Remain voters as possible” amid speculation that thousands of those prevented from registering were younger people who are thought to be more likely to back EU membership.
In a message to supporters, Mr Elliott urged them to get Brexit-backing friends and family to apply for a vote, with the message: “Don’t let the Government skew the result of the referendum - make sure you and your friends are all registered today.”
Downing Street said 214,000 people were trying to use the www.gov.uk/register-to-vote website between 9pm and 10pm on Tuesday, but it was not known how many were prevented by the system overload from registering before the midnight deadline.
Amid demands from senior figures including Labour leader Jeremy Corbyn, Liberal Democrat chief Tim Farron and Green leader Natalie Bennett for a deadline extension, ministers went into urgent talks with the Electoral Commission over possible legislative action to allow late applicants to vote.
And Mr Cameron urged people to keep on registering while the situation was sorted out.
In a message on Twitter, he said: “If you aren’t registered and you want to vote in this EU referendum you should continue to register at gov.uk/register-to-vote.
“Huge demand caused system overload last night. Urgent discussions with Electoral Commission to try to ensure people who register can vote.”
The chairman of the Commons Public Administration and Constitutional Affairs Committee, Bernard Jenkin - a leading Leave campaigner - warned the Government was acting on the “cusp of legality” and may be faced by a judicial review if the referendum result was close.
He said the law required the register to be finished and published six working days before the poll, with a further five days in which new entrants on the register can be challenged, when there are just 14 working days before the vote.
“We are on the cusp of legality here. We are now outside the legal framework by keeping the registration open today,” Mr Jenkin told BBC Radio 4’s World At One.
“If they were to extend the registration for a period longer than this, they are really opening the possibility of a judicial review of the result if it is very close and people feel they have got a cause.
“We are not a banana republic. We shouldn’t be making up the rules for our elections as we go along. This is a shambles.”
Leading Brexit campaigner Michael Gove said: “These are complex legal waters and I do not want to presume definitively without having looked at the law. But in my heart is a desire to ensure that everyone possible can be given the vote.
“The more people who vote the better. This is a lifetime-defining decision. I would like to see everyone who possibly can and who is entitled to vote play a part in this.”
Asked if he feared a higher turnout could damage the chances of a Leave vote, he said: “I am confident that the more people who hear our message, the more will vote to leave.”
Alex Robertson, director of communications at the Electoral Commission, said: “No-one should miss out on voting in this historic referendum because of the problem with the Government’s registration website last night. We said this morning that legislation should be introduced to extend the registration deadline and we’re pleased the Government will now be making this change.
“We are urging everyone who is not already registered to vote to take this last chance to do so before the end of Thursday.”
Shadow voter registration minister Gloria De Piero said: “We welcome the Government’s decision to listen to Labour’s call to extend the deadline for voter registration.
“Following the chaos of last night, tens of thousands of people will be relieved that they will have a say in this crucial vote.
“We urge everyone who thinks they are not on the register to get back on that website and have their say.”