Keogh is alleged to have passed the memo to O'Connor in May 2004, who in turn placed it in a file he handed to his boss, Tony Clarke, then a legislator who had voted against Britain's decision to join the US-led invasion of Iraq in 2003.
Britain's Daily Mirror newspaper previously had reported that the memo noted that Blair had argued against Bush's suggestion of bombing Al Jazeera's headquarters in Doha, Qatar.
The Daily Mirror said its sources disagreed on whether Bush's suggestion was serious.
'Outlandish and inconceivable'
Blair has said he had no information about any proposed US action against Al Jazeera, and the White House called the claims "outlandish and inconceivable".
David Perry, the prosecutor, did not mention the claim in court on Wednesday, but said jurors would see the document during parts of the hearing, closed to the public, because of the sensitivity of the contents.
The document - marked "Secret-Personal" - was intended to be restricted to senior officials and was written by a Blair adviser, he said.
Blair's staff circulated the memo to a range of military and political officials in London, Washington, the United Nations and Iraq - including Britain's MI6 intelligence agency.
Keogh and O'Connor had put the lives of troops in Iraq at risk because the memo contained defence data, Perry said.
Bush and Blair met in Washington on April 16, 2004, while the Coalition Provisional Authority was acting as administrator in post-war Iraq and "against the background of the insurgency."
"We live in a democratic society, not the Wild West," Perry told the court.
"It is not for people to decide they are going to be the sheriff in town."
Keogh worked at a government communications bunker handling sensitive documents and intelligence, Perry said.
The unit relayed information to diplomats overseas via encrypted or secure methods.
Perry said Keogh received a faxed copy of the memo to send on to official, but duplicated it unlawfully before doing so and later passed the document to O'Connor.
Clarke, who is no longer a lawmaker, alerted authorities when he discovered the memo among paperwork from O'Connor.
Britain's foreign office said Keogh is currently suspended pending the outcome of the case.
Both defendants deny offences under the Official Secrets Act.
Keogh denies two charges of making a damaging disclosure of part of a government document. O'Connor denies two similar charges.