The contractor will restore the Victorian glasshouse, built in 1863, and repair and replace ventilation and irrigation systems.
It will also improve facilities and accessibility for visitors and staff at the largest remaining Victorian Glasshouse in the world.
The £24m building contract is part of a larger £36m restoration project.
The Temperate House is and houses Kew’s collection of temperate-zone plants. Kew Gardens has spent the past year moving large, mature plants and growing new ones to enter the restored glasshouse.
ISG’s Southern regional managing director Stuart Deverill said: “The Royal Botanic Gardens at Kew has a global reputation for plant research and conservation and to play a role in supporting that important work through this refurbishment project is a real privilege.
“The Temperate House is the largest remaining Victorian glasshouse and we are delighted to be working to protect this stunning building, while creating an environment where plants, education and tourism can flourish.”
Work is due to complete in spring 2017, then there will be a year of replanting work, with plans for a formal re-opening in May 2018.
RBG Kew Temperate House project director Andrew Williams said: “We are delighted that work can now commence on this vital restoration of one of the nation’s most important historic horticultural buildings, built to protect, display and enable education and research using a living collection of plants from the temperate regions.”
ISG also delivered restoration of the Grade I listed Royal Festival Hall and the Grade II Vaughan’s Mansion in Shrewsbury, and carried out the £16m refurbishment of the Grade I listed Royal Institution of Great Britain.