The gag order from Attorney-General Lord Goldsmith came nearly 24 hours after the paper published details of what it said was a transcript of talks between Bush and the British Prime Minister Tony Blair.
In those talks, which took place during the prime minister's April 2004 visit to Washington, Blair is said to have talked Bush out of launching "military action" on the television channel's headquarters in Doha, Qatar.
"No 10 did nothing to stop us publishing our front page exclusive yesterday (Tuesday)," the Daily Mirror said on Wednesday, referring to the British prime minister's office.
But the attorney-general warned that publication of any further details from the document would be a breach of the Official Secrets Act.
He threatened an immediate High Court injunction unless the newspaper confirmed it would not publish further details.
"We have essentially agreed to comply," the paper reported.
"We made No 10 fully aware of the intention to publish and were given 'no comment' officially or unofficially, Daily Mirror Editor Richard Wallace was quoted as saying.
"Suddenly 24 hours later we are threatened under Section 5 [of the Official Secrets Act]."
According to Britain's Guardian newspaper, it is the first time that the Blair government has threatened to prosecute a newspaper for publishing the contents of leaked government documents.
Scott McClellan described the
Mirror's report as 'outlandish'
The White House has dismissed the Daily Mirror report, calling it "outlandish".
"We are not going to dignify something so outlandish and inconceivable with a response," White House spokesman Scott McClellan said on Tuesday.
Aljazeera itself, whose coverage of the war in Iraq has been criticised by the US, says it is also investigating the report.
"If the report is correct then this would be both shocking and worrisome not only to Aljazeera but to media organisations across the world," the station said in a statement.
Following the Mirror's report there have been calls to release the transcript.
"If true, then this underlines the desperation of the Bush administration," said Liberal Democrat foreign affairs spokesman Sir Menzies Campbell.
"If the report is correct then this would be both shocking and worrisome not only to Aljazeera but to media organisations across the world"
"On this occasion, the prime minister may have been successful in averting political disaster, but it shows how dangerous his relationship with President Bush has been."
The Mirror on Tuesday quoted a source as saying: "The memo is explosive and hugely damaging to Bush.
"He made clear he wanted to bomb Aljazeera in Qatar and elsewhere. Blair replied that would cause a big problem.
"There's no doubt what Bush wanted to do - and no doubt Blair didn't want him to do it."
The threat by Bush also "casts fresh doubt on claims that other attacks on Aljazeera were accidents", the Mirror said in its report on Tuesday.
Reporter Tariq Ayub died in a US
strike on Aljazeera's Iraq bureau
It cited the 2001 direct hit on the channel's Kabul office in Afghanistan.
In November 2001, Aljazeera's office in Kabul was destroyed by a US missile. None of the crew was at the office at the time.
US officials said they believed the target was a "terrorist" site and did not know it was Aljazeera's office.
In April 2003, an Aljazeera journalist, Tariq Ayub, died when its Baghdad office was struck during a US bombing campaign.
In its statement on Tuesday, Aljazeera said that if the Mirror's report was true, it would "cast serious doubts in regard to the US administration's version of previous incidents involving Aljazeera's journalists and offices."
A British civil servant has been charged under the Official Secrets Act for allegedly leaking the government memo.
The Daily Mirror said the memo, stamped "Top Secret", turned up last year at the Northampton office of then Labour MP Tony Clarke.
Civil servant David Keogh, 49, is now accused of passing the memo to Leo O'Connor, who once worked for Clarke.
Both Keogh and O'Connor are due to appear in court next week on charges under the Act.
Clarke, who opposed the invasion of Iraq and who lost his seat at the last election, returned the memo to Downing Street.