It will work alongside Atkins, which will design the scheme, forming a key part of the Friargate development at the railway station.
The 100m wide bridge deck across the ring road will connect the Friargate scheme to the city centre by removing the raised roundabout by junction 6 to create a public boulevard linking to Greyfriars Green.
The Council will be working closely with Costain and Atkins over the next few months as the detailed designs are put in place.
Work is expected to start early next year – with completion set for June 2015.
Cllr Rachel Lancaster, Cabinet Member for Public Services, said: “The construction of the bridge deck and the Friargate development that follows will kickstart the regeneration of our city.
“The bridge work is being fully funded from a Regional Growth Fund grant and I’m pleased that there was so much interest in undertaking both the design and build work.
“This is the perfect opportunity to break through the ring road ensuring greater pedestrian access from the city centre to the railway station, without disrupting the flow of traffic on the ring road which at junction 6 runs underneath the junction.
“Once completed, the bridge deck and new boulevard will make up one of the most attractive and impressive walking routes into a city centre from a railway station of any city in the UK.”