The Government-funded Higgs Centre for Innovation will be built at the city’s Royal Observatory site.
In partnership with the University of Edinburgh, the new centre will focus on space and big data technologies.
Graham won the job with a £6.4m bid. The total scheme is expected to cost around £12m allowing for fit-out with research equipment.
Consultant CPC is providing project and cost management consultancy.
Prof John Womersley, chief executive of the STFC, said: “Particle physics, astronomy and space science all address one of the biggest questions in science, what is the universe made of.
“The Higgs Centre for Innovation will significantly increase the positive impact that arises from fundamental research like this, both in job creation and economic opportunities and growth in the UK.”