The approval allows the Trust to complete negotiations to sign a contract with Laing O’Rourke in the summer.
In March the Trust secured the £70m loan for the refurbishment from the Department of Health.
The project includes a new ward block to be built on the Terrell Street site, refurbishing the Queens Building and converting wards in the old King Edward building.
The new ward block was first discussed five years ago and should be completed by 2016.
This is the second full business case that University Hospitals Bristol NHS Foundation Trust has signed up to in as many months.
In February, the Board agreed the business case to centralise specialist children’s services at the Bristol Royal Hospital for Children.
Paediatric burns, orthopaedics, plastic surgery and neurosciences services are due to move from Frenchay Hospital in 2014.
Services will be housed in an extension to the Bristol Royal Hospital for Children which will link to the new ward block for the BRI.
Last week Bristol City Council granted planning permission for the extension to the Children’s Hospital.