He wants Lord Falconer of Thoroton to oversee the pounds 758m Greenwich exhibition, which Mr Mandelson had made his personal crusade.
The appointment will be seen as a sign that the Prime Minister will not be diverted from his New Labour programme of reforms by John Prescott, who last week cheered the Old Labour rebels by calling for the Government to return to "traditional Labour values".
Other ministers are reported to have called for a shift in direction following the resignation of Mr Mandelson. Aides to Mr Prescott say the Deputy Prime Minister's allies in Government have been enthusiastically suggesting that he take on responsibility for the Dome.
But Government sources believe the Secretary of State for the Environment, Transport and the Regions, who is a keen supporter of the Dome, is unlikely to want to take on the extra portfolio because he is too busy.
The choice of Charlie Falconer, a key Blair ally, who was made a peer by the Prime Minister on entering office, will be partly motivated by his contacts across Government.
The former Solicitor General sits on 16 Cabinet committees and ministerial groups and is the right-hand man to the Government's enforcer, Jack Cunningham. He is regarded as Mr Blair's "eyes and ears" in government and has known the Prime Minister since they were students.
Since its inception, the Dome has been beset by controversy. It has 14 themed zones, including the body zone, which features a giant abstract sculpture of a reclining female in a "gentle embrace" with her male partner.
Chris Smith, the Secretary of State for Culture, Media and Sport, has taken on responsibility for the Dome in a caretaker capacity since Mr Mandelson resigned from the Government, but he is responsible for the body which funds millennium projects and there is a potential conflict of interest.The Prime Minister is expected decide who will take on the Dome later this month when Parliament returns after the Christmas recess.
The new Minister will be responsible for boosting sponsorship; the project has not yet gained its full quota of sponsors, although it is backed by Boots the Chemist and McDonalds, which is likely to open several outlets on the site.
The project also faces doubts about whether the train service now being built will be ready in time to transport visitors to the exhibition site when it opens on 31 December 1999.
London Underground's Jubilee Line Extension is supposed to carry two in five visitors to and from the Dome.Reuse content