The University announced Wates had won the contract 12 months ago ahead of BAM, Kier and Mansell.
But the contractor failed to agree final terms on the job, causing the university to approach rival bidders.
BAM faces several challenges, building on a steeply sloping site that is constrained by other buildings.
The contractor will also oversee the fit-out of the centre, including laboratories and complex services.
Building information modelling has been used from the outset, and an early relationship was established with mechanical and electrical contractor NG Bailey.
The new centre will bring together mathematicians, scientists and engineers in a bid to revolutionise the diagnosis and treatment of a wide range of human, animal and plant diseases.
It will consist of two linked buildings on a split level site, with a tower block nine storeys high and a slab block reaching six storeys, three storeys being basement levels with significant excavations and temporary works.
An Excellent rating is being targeted through environmental initiative BREEAM, with solar panels and high levels of insulation. A combined heat and power plant will allow the centre to create its own energy.
The project enters its main construction phase this summer, with sectional completion in March 2016 and overall completion scheduled for May 2016.
The architect is Hawkins Brown, Civil & Structural engineers are AECOM and Services engineers are Hoare Lea.
BAM is also building the new £11m South Devon University Technical College (UTC), in which the University is a partner.
The contractor has worked on a number of schemes in Exeter and Plymouth, including the Royal Albert Memorial Museum, Montgomery School, Exeter Business school, Marine academy; All Saints Academy; the Roland Levinsky Building at Plymouth University; Plymouth UTC; and Plymouth Theatre Royal.
This has allowed it to build up an extensive list of suppliers in the county, a major advantage in securing this latest scheme.
BAM construction director Graham Kingdon said: “We have worked hard over a number of years to create our supply chain in the South West. We now have a number of trusted, quality companies we can ask to price work for us.
“This allows us to find the best suppliers at the best prices and ensure we can deliver world class facilities in Devon and Cornwall even at times of economic hardship.”