"Having consulted political colleagues, family and friends, I have reached the conclusion that I should not put my hat in the ring," he said.
Mr Campbell, 58, is regarded as the strongest supporter of the "project" for closer links with Tony Blair's Government. Although Mr Kennedy, as the front-runner, is trying to position himself in the middle, to avoid upsetting supporters and opponents of the closer links, he will gain the most from Mr Campbell's decision.
Some of the party grandees who would have supported Mr Campbell, the Liberal Democrats' defence spokesman, including Lord Jenkins of Hillhead have privately told colleagues they will switch to Mr Kennedy.
Mr Campbell's move led to calls for some of the other candidates to drop out to make the election a clear choice between Mr Kennedy, who supports close links with Labour, and one of the radical opponents of the "project", who oppose moves which could lead to coalition in a future Labour Government.
Mr Campbell, who is a member of the joint constitutional committee with the Cabinet, was closely associated with the "project" for closer ties with the Government, leading possibly to coalition.
He jointly announced plans for a defence pillar in Europe with Robin Cook, the Foreign Secretary.
His departure makes it more likely that Mr Kennedy, another pro-project candidate, will win. There are now expected to be four firm candidates challenging him as the front-runner for the leadership, but there were calls for some of the critics of closer links with Labour to stand down, to avoid splitting the vote.
Candidates who oppose closer links with Labour are Jackie Ballard, David Rendel and Simon Hughes.
A fourth candidate, Don Foster, supports the "project" and is positioning himself as a candidate who is more enthusiastic than Mr Kennedy for an alliance with Labour in government.
A supporter of Mr Foster said: "This now means Foster is the pro-project candidate and others will swing behind Foster now that Menzies is out of the running. Foster is the main challenger to Kennedy."
They concede that Mr Kennedy starts in front but claim there will be momentum behind the Foster campaign and they are claiming it will be between Mr Kennedy and Mr Foster.
The anti-project campaigners are "all fighting among themselves" and cannot decide who is going to run, said the Foster supporter.
Mr Foster's camp is convinced the majority of the party are in favour of the closer ties with Labour and the real battle therefore will be between Mr Foster and Mr Kennedy.
But a Rendel supporter rejected those claims: "Jackie, Simon and Don should be thinking about their position. Menzies' departure shows that a pro-project candidate cannot get sufficient support and that David now has the best chance of winning."
Ms Ballard is highly unlikely to drop out to allow Mr Rendel a free run against Mr Kennedy.
She is a doughty campaigner, and will base her campaign manifesto on Liberal issues including opposition to welfare cuts, which is one of the reasons she believes the Liberal Democrats must remain free to criticise the Government's policies.
Mr Rendel is hoping to secure his power base among Liberal Democrat councillors for the ballot of the entire membership, and is also unlikely to pull out.
Paddy Ashdown, whose decision to retire from the leadership caused the contest, has told the candidates not to start campaigning until after the European elections on 10 June are over.
However, all the candidates have active campaign teams at Westminster.Reuse content