Mr Morgan, who earlier met Labour AMs, told a media conference: "It is never a right time to go but this is the nearest in terms of good governance, combating swine flu, 'settling' the budget, and combating the recession."
"I have loved this job for the past nine-and-a-half years, I still love this job."
Mr Morgan added that he did not want to be "greedy" for power and to be forced out as Margaret Thatcher had been.
He told a story about his six-year-old grandson who was asked by his friends whether his grandfather was standing down as first football manager or first rugby manager of Wales and his grandchild replied, "Both".
Mr Morgan said that was true and it had been a "burden".
He said his biggest success was the new Scandinavian type education curriculum for three to seven-year-olds (the foundation phase).
First Minister Rhodri Morgan said the time was right to go after nearly a decade in the role
"Ten years ago we would not have had the power or the confidence".
But he said for the public the biggest success would be free bus passes for pensioners, while his biggest disappointment was the "savagery of the recession" which hit the Welsh economy.
Mr Morgan arrived for the meeting with his party's AMs just after 1230 BST and a formal announcement was made afterwards.
The timetable of his departure centres on discussions on the assembly government later this year.
On 8 December, the budget for 2010-2011 is due to be voted on by AMs and Mr Morgan wanted to stay in post until that date.
Prime Minister Gordon Brown said his achievement had been "historic".
He said: "I think all of us who know Rhodri understand his devotion and passion for serving the people of Wales.
"For a decade he has led Wales with boundless energy and commitment - improving the public services, economy and prospects of people across Wales.
"It is in no small part because of Rhodri's leadership that devolution to Wales continues to be so successful. He has been a great support to me and will depart with my very grateful thanks and that of the whole Cabinet and Labour Party."
Welsh Secretary Peter Hain added that Mr Morgan was "a great Welsh political leader with a unique popular touch".
He has the rare ability to be equally at home whether it be meeting world leaders or talking rugby over a pint in a pub
Welsh Secretary Peter Hain
Mr Hain said: "Admired and respected for his courageous, intelligent and sometimes quirky leadership, he has the rare ability to be equally at home whether it be meeting world leaders or talking rugby over a pint in a pub."
"A passionate Welshman, but also a passionate Briton," he said.
"Rhodri will depart at the top of his game, leaving big shoes to fill as the father of Welsh devolution."
The election to find a successor would not start until a timetable has been set by the Welsh executive committee, said Garry Owen, chair of Welsh Labour.
It is widely expected the contenders to take over as leader of the Labour group in the assembly - and first minister - will include three leading AMs - Counsel General Carwyn Jones, Merthyr Tydfil and Rhymney AM Huw Lewis, and Health Minister Edwina Hart.
Mr Morgan said he would not be endorsing a candidate in the forthcoming leadership election.
'Time for woodcarving'
"I will be neutral... It will be the first Labour campaign in the last 48 years that I have not been involved in," he added.
But would continue to campaign for the party but hoped standing down would give him "more time on the allotment and time for woodcarving".
Welsh Liberal Democrat leader Kirsty Williams said: "Nobody can detract from Rhodri Morgan's commitment to Wales and to devolution.
"Rhodri has always been a kind and extremely personable colleague, a real people-person. I'll miss our political sparring and even his rugby anecdotes in the chamber but it really is time for a fresh start, fresh faces and fresh ideas in Cardiff Bay."
Ms Williams claimed that Morgan had "single-handedly held Welsh Labour together for nearly a decade" and whoever followed him would face "huge challenges".
Nick Bourne, leader of the Conservatives in the assembly, called for a quick resolution of the battle to succeed Mr Morgan.
"No-one can question his commitment to Wales and the assembly during his 10 years as first minister, or the important role he has played in Welsh political life," said Mr Bourne.
"What is important now is that at a time of economic crisis the assembly has leadership, vision, and direction."
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