English Heritage chief executive Simon Thurley said the road scheme, which will be built by Balfour Beatty and Costain for the Highways Agency, will be delayed by a public inquiry.
Thurley said the two projects would remove all signs of the 20th century from the site. He said: "This is the end of a 16-year quest to find a solution to Stonehenge. We are very excited."
The visitors' centre will receive £26m from the National Lottery, £10m from the government and £11.7m from English Heritage. The other £9.3m is hoped to come from a fund-raising drive, which will be launched by English Heritage this year.
Thurley described the design as "very subtle".
This is the end of a 16-year quest to find a solution to Stonehenge
Simon Thurley, head of English Heritage
He said: "The intention was not to make a monument in itself. This is a building that naturally couches itself in the countryside – it's very discreet, which is deliberate."
The visitors' centre will be outside the World Heritage site and out of sight of the stones, 3 km from the present concrete visitors' centre. A customised bus service will shuttle visitors to within 1.4 km of the site. The final approach will be covered on foot, allowing visitors to appreciate the site within open countryside.
Australian practice Denton Corker Marshall won the competition to design the centre in April last year, beating off Michael Hopkins and Partners and Allford Hall Monaghan Morris. It will be the firm's first UK project.
Barrie Marshall, the founder of the firm, said the key objective was to return the stones to their natural setting. He said: "The objective is to redefine their relationship with the landscape without cluttering it up with other buildings and roads."