The budget for the hospital was $1.3 billion, but SNC-Lavalin ran up an extra $172 million.
The provincial government and SNC-Lavalin have gone to a mediator to negotiate the payment.
The engineering giant is asking the government to cover the extra costs, but Health Minister Gaétan Barrette said the government is not going to pay a penny more than the original price.
The hospital is scheduled to open by the end of April 2015.
Now that the keys have been officially handed over to hospital management, the next step will be moving in equipment and preparing for patients.
Barrette said the hospital will provide "maximum quality care."