It won the job more than a year ago after re-egineering the structure to cut costs by one-fifth.
The bridge project has suffered several delays, the most serious when ministers said that plans to replace the current temporary structure were too costly.
This prompted a cost-saving exercise that sliced around £6m from the estimated project cost by redesigning the foundations and transferring extra risk to Costain.
Carbon steel will be used instead of weathering steel, while further savings will be made by reusing excavated material.
The design drawn up by Atkins avoids placing piers in the water to provide unrestricted views along the river and improving navigation for boats.
The Department of Transport has agreed to spend £23.8m with the rest of the cash coming from Surrey County Council.
The council has be waiting for 57 years to replace two temporary bridges erected to replaced the original bridge damaged during the Second World War.