A dozen cabinet ministers then objected to the project, but eventually decided to go ahead with it, even after admitting they could not blame the previous Conservative Government if it went wrong.
Leaked documents show that a number of senior ministers, including the Chancellor, Gordon Brown, had advised against it, according to The Mail on Sunday.
But Mr Blair is said to have told them the scheme had to go ahead, and none of the ministers wanted to defy him.
The report comes just days after the publication of a damning report by the National Audit Office into the running of the Greenwich attraction.
Among the criticisms heaped on the Dome in the report were that it was risky, mismanaged and financially weak before it even opened.
The Mail on Sunday has reproduced what it says is the first full record of a Cabinet meeting, which is usually suppressed for 30 years under laws governing Whitehall secrets.
At the meeting, held in June 1997, shortly after Labour came to power, Mr Blair is quoted as saying: "It is an important decision and we should proceed in principle."
'Dishonesty'
The only alternative would be to write off "huge sums" of public money which had already been committed to the scheme, he is said to have added.
According to the paper, Mr Blair then left the meeting to attend a memorial service, leaving his deputy, John Prescott, in the chair.
A dozen cabinet ministers then objected to the project, but eventually decided to go ahead with it, even after admitting they could not blame the previous Conservative Government if it went wrong.
The newspaper quotes Chancellor Gordon Brown as voicing a "series of worries" about the Dome, and the Education Secretary, David Blunkett, saying he was "deeply against" it.
Chief Whip Ann Taylor suggested the money for the project could be spent on schools and hospitals instead, and the International Development secretary, Clare Short, warned of a "political disaster".
The report says the cabinet agreed that despite their misgivings, if Mr Blair backed the Dome, they should rubberstamp it.
'Human shield'
Foreign Secretary Robin Cook is quoted as saying "If Tony has made a decision, we'll all have to support it", before going on to speak of his own "grave" anxieties about the project.
Shadow culture secretary Peter Ainsworth said the leaked documents exposed "the dishonesty at the heart of this government on this issue".
He said the report showed the hypocrisy of Labour's attempts to blame Tory mistakes for the Dome's failings.
He also claimed Dome Minister Lord Falconer was not being allowed to resign because it would expose Tony Blair to the "firing line".
He told GMTV's Sunday Programme that the Prime Minister was using his former flatmate as "a very useful human shield" to protect himself from criticism.
"Lord Falconer is terribly important to Tony Blair because if he goes then Tony is in the firing line," he said
"It is an outrage that Lord Falconer has not resigned. If Lord Falconer does not resign it will leave a long-lasting and bitter taste in the public's mouth over this whole affair."
Chancellor Gordon Brown told BBC1's Breakfast with Frost programme he had never seen the report but it did not effect the two central issues at the heart of the row over the Dome's failings.
The Cabinet shared collective responsibility for the decision to go ahead whatever their reservations beforehand, he said.
Mr Brown said Conservatives were also to blame for their part in setting up the project and producing unrealistic targets for visitor figures.
Downing Street has refused to confirm whether the account of the meeting is accurate, or reveal if attempts would be made to find the source of the leak.