Mr Brown said he had asked the Labour Party to set in train the formal mechanism to replace him, after insisting he was confident an administration could be formed with Lib Dem allies to command a majority in the Commons.
The Prime Minister said: "The reason that we have a hung parliament is that no single party and no single leader was able to win the full support of the country.
"As leader of my party I must accept that as a judgment on me. I therefore intend to ask the Labour Party to set in train the processes needed for its own leadership election.
"I would hope that it would be completed in time for the new leader to be in post by the time of the Labour Party conference.
"I will play no part in that contest, I will back no individual candidate."
He made his announcement in Downing Street after another day of Westminster drama, when hopes had seemed high of Tory leader David Cameron and the Lib Dems' Nick Clegg doing a deal.
Mr Brown clearly felt, after Mr Clegg had formally requested opening negotiations with Labour, that his party's only chance of succeeding in talks to head off Mr Cameron moving into No 10 was to quit.
The Prime Minister said his aim was to ensure a "stable, strong and principled government" was formed.
The Liberal Democrats thought they should first talk with the Conservatives, he said.
"Mr Clegg has just informed me that while he intends to continue his dialogue that he has begun with the Conservatives, he now wishes also to take forward formal discussions with the Labour Party.
"I believe it is sensible and it's in the national interest to respond positively.
"The Cabinet will meet soon. A formal policy negotiating process is being established under the arrangements made by the Cabinet Secretary similar to the negotiations between other parties.
"The first priority should be an agreed deficit reduction plan to support economic growth and a return to full employment."
Mr Brown's resignation will be seen as the clearest possible signal that Mr Clegg made plain his party could not countenance a deal with Labour while he was still at the helm.
The premier's decision to stand down as leader was swiftly welcomed by some Labour MPs.
John Mann, who became the first backbencher to call for him to go on Saturday, hailed it as a "wise and brave" decision.
He said the party now needed to "regroup" to fight the Tories in Opposition.
Slough MP Fiona Mactaggart, who has criticised Mr Brown's leadership in the past, said Mr Brown had done the "right thing".
"It is a recognition that Labour lost the election which I think has not been made clear until this point," she told BBC Radio 4's PM programme.
Mr Brown's announcement came after 90 minutes of further talks earlier today at the Cabinet Office between the Tory and Lib Dem negotiating teams, and was clearly designed to wrong-foot Mr Cameron before he met his own MPs in the Commons.
Tory sources revealed that Mr Clegg and Mr Cameron also met face-to-face again this morning while the negotiating teams were locked in Whitehall talks.
The latest 30-minute conversation, which followed a half hour telephone chat earlier in the day, "went well", the source said.
Mr Clegg then gave a progress report to his own backbenchers and it was after that meeting in Westminster that Mr Brown delivered his statement outside No 10.
Lib Dem negotiator David Laws said after his party's MPs' meeting that it was Labour who were seeking negotiations.
After the Lib Dems MPs' briefing, he said: "This discussion is dependent not only on the Liberal Democrat party but also on the proposals and discussions that are on-going with the Conservative Party and the representations that frankly are being made by the Labour Party."
But Mr Brown told reporters the Lib Dems had now asked for official talks, and summoned his Cabinet for 6pm.
As they trooped through the door of No 10, many would be wondering who they would back as the premier's successor - regardless of the outcome of the power-sharing talks.
Deputy leader Harriet Harman, Home Secretary Alan Johnson, Children's Secretary Ed Balls, Foreign Secretary David Miliband and his brother Energy Secretary Ed Miliband have all been tipped as possible replacements.
Meanwhile, Tory MPs were digesting the Prime Minister's announcement as they quizzed Mr Cameron at their private Westminster meeting, on how far he was prepared to go to win Lib Dem support to gain the keys to No 10.
Cabinet minister Douglas Alexander - Labour's general election co-ordinator - said Mr Brown had decided to step down last week.
"He made a judgment shortly after the election results that this was the right course of action," he told Sky News.
"He is very keen to ensure that he meets his constitutional obligations which is to ensure that a government is formed.
"If that government involves the Labour Party then, of course, he would be leading the Labour Party and would be Prime Minister but he has also made clear his determination to assume responsibility for the result on Thursday evening and, in time, to demit office."
He insisted Mr Brown's intervention was "dignified rather than desperate".
"What we have heard from Gordon Brown today was a statesmanlike statement in which he clarified his own position and his desire to facilitate those discussions being taken forward."
Mr Cameron made no comment about the latest developments as he arrived at a committee room in the House of Commons to address his MPs.
There was loud applause and banging of desks as he entered the room shortly before 6pm.
The Liberal Democrat leader later hailed Mr Brown's announcement as possibly an "important element" in the "smooth transition" to a stable government.
Mr Clegg said Mr Brown had made an "important" announcement.
"It must have been a very difficult thing for him to say personally. But I think he's taken it in the national interest and I think his announcement could be an important element in the smooth transition towards a stable government that people deserve - without prejudicing or predicting what the outcome of the talks will be between ourselves and the Labour party."
Transport Secretary Lord Adonis, a former Liberal Democrat who is one of Labour's team for negotiations with the third party, rejected suggestions that Mr Brown's departure was prompted by demands from Mr Clegg for him to go as part of any deal.
Lord Adonis told BBC Radio 4's PM: "Gordon's decision to hand on the Labour leadership is a decision he reached himself after considering the results of the election. It had nothing whatever to do with these talks with the Liberal Democrats.
"Gordon's view is that after the results of the election, it was right that a new leader should take office."
Lord Adonis added: "I hope that these discussions with the Liberal Democrats are successful and I very much hope and - though I can't speak for them - I think it is possible for us to form a coalition government.
"So the prime quality we will need in a new Labour leader is fitness to hold the office of Prime Minister. I think Labour Party members, if we do remain in government, will be looking very carefully at which is the leader who is fit to be Prime Minister."
Mr Brown's dramatic announcement came shortly after a two-and-a-half hour meeting of Liberal Democrat MPs demanded "clarification" from the Tories of the proposals being put forward for co-operation between the parties, including on the key issue of electoral reform.
The party's MPs also called for further information about plans regarding education funding and fairer taxes.
Lib Dem negotiator David Laws said the party had agreed Mr Clegg would also continue to listen to representations from Labour while clarification was sought about the offer from the Tories.
The MPs agreed that the "central priority must be to form a strong and stable government in the national interest" and that reducing the deficit should be "at the heart of any agreement", said Mr Laws.
The Lib Dem frontbencher said the MPs agreed that "very good progress" had been made in the talks with Tories, which have taken place over the past four days. Both sides emerged from the latest negotiations this morning saying they had made "further progress".
But Mr Laws added: "They have also asked for clarification of details in relation to education funding, fair taxes and in relation to issues in regard to voting reform and progress on that issue."
Mr Cameron briefed his shadow cabinet on the state of the negotiations, and was this evening addressing Tory MPs at a crunch gathering of the backbench 1922 Committee in Westminster.
Tory sources revealed that Mr Clegg and Mr Cameron also met face-to-face again this morning while the negotiating teams were locked in talks at the Cabinet Office.
The latest 30-minute conversation, which followed a half hour telephone chat earlier in the day, "went well", the source said.
The Lib Dems' ruling federal executive, which must approve any deal which affects the party's political independence, was meeting this evening and a further meeting of the Lib Dem MPs is expected to be held later in the day.
Federal executive members arriving for the meeting at the Work Foundation thinktank had to make their way through a crowd of protesters - many dressed in purple - chanting "no deal without fair votes" and "make votes count".
Lib Dem chief executive Chris Fox told the demonstrators: "I understand your concerns, we share them.
"We are working as hard as we can within the electoral system we have to bring about the change that you all, and we all want."
A Labour Party spokesman said: "Gordon Brown has always felt the most important priorities for the country are securing the recovery and changing our politics for good. Gordon Brown's words today were the mark of a man who has always put his country first.
"The Labour Party's National Executive Committee will meet in the coming days to determine the procedures for a leadership election."
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