Other consortium members include consultant engineer Arup and M&E engineer BDSP.
Hart was praised by former Welsh secretary Ron Davies, now an assembly member, for her "courage" in rehiring Lord Rogers.
Hart said she was confident that the contract would keep prices under control. She said she hoped that it could be signed in early June, when the contractors put forward their price.
"We are into a different ball game," said Hart, in comparison with the previous deal with Lord Rogers. Last time the scheme was criticised by auditors for its cost control procedures.
Mrs Hart refused to give any estimation of the price to the assembly's Conservative group – which remains opposed to the project – beyond criticising "some very wild exaggerations".