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John Scarlett, chairman of the joint intelligence committee (JIC), also "assented" that Saddam had no such weapons, says Cook.
His revelations, taken from a diary that he kept as a senior minister during the months leading up to war, are published today in The Sunday Times. They shatter the case for war put forward by the government that Iraq presented "a real and present danger" to Britain.
Cook, who resigned shortly before the invasion of Iraq, also reveals there was a near mutiny in the cabinet, triggered by David Blunkett, the home secretary, when it first discussed military action against Iraq.
The prime minister ignored the "large number of ministers who spoke up against the war", according to Cook. He also "deliberately crafted a suggestive phrasing" to mislead the public into thinking there was a link between Iraq and Al-Qaeda, and he did not want United Nations weapons inspections to be successful, writes the former cabinet minister.
Cook suggests that the government misled the House of Commons and asked MPs to vote for war on a "false prospectus".
He also reveals that Blair earlier gave President Bill Clinton a private assurance that he would support him in military action in Iraq if action in the UN failed "and it would certainly have been in line with his previous practice if he had given President Bush a private assurance of British support".
Cook's long-awaited diaries, published in book form as Point of Departure, are the first memoir of any member of Blair's cabinet. His disclosures are likely to lead to renewed calls for a judicial inquiry into the legitimacy of the war.
The Hutton inquiry into the death of Dr David Kelly has dealt only with the question of what the government believed ahead of publication of its Iraq dossier in September 2002 and whether Downing Street hardened intelligence reports to make the threat from Saddam seem more compelling.
Cook today opens a new controversy. He says that just days before sending troops into action, Blair no longer believed Saddam had weapons of mass destruction ready for firing within 45 minutes, the claim the prime minister had repeatedly made when arguing the case for war.
Cook reveals that on February 20 this year he was given a briefing by Scarlett. "The presentation was impressive in its integrity and shorn of the political slant with which No 10 encumbers any intelligence assessment," Cook writes in his diary. "My conclusion at the end of an hour is that Saddam probably does not have weapons of mass destruction in the sense of weapons that could be used against large-scale civilian targets."
Two weeks later, on March 5, Cook saw Blair. At the time the government was still trying to get a fresh UN resolution and Cook was still in government as leader of the Commons.
Cook writes: "The most revealing exchange came when we talked about Saddam's arsenal. I told him, 'It's clear from the private briefing I have had that Saddam has no weapons of mass destruction in a sense of weapons that could strike at strategic cities. But he probably does have several thousand battlefield chemical munitions. Do you never worry that he might use them against British troops?'
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