The original budget for the project was £170m.
But the government confirmed a revised cost of £286m as Lendlease started work on the new HMP Glen Parva.
Prisons Minister Lucy Frazer said the scheme will be built using “eco-friendly, cutting-edge techniques.”
Parts of the prison will be built off site while most of the ground materials will be recycled, with 75,000m3 of existing material due to be re-used.
Work is expected to be completed by spring 2023.
Frazer said: “As well as rehabilitating prisoners and keeping the public safer, the prison will be a major boost to the local economy.
“Hundreds of jobs will be created during and after construction, sending a clear signal that this government is committed to supporting communities and infrastructure across the country.”
Simon Gorski, Lendlease’s Managing Director for Construction in Europe, said: “We have a proven track record of delivering projects that can deliver economic, social and environmental benefits for the communities where we work.
“Our partners trust that we will deliver long lasting value and we will be entirely focussed on delivering the very highest standard of building.”