The museum, a former church located next to Lambeth Palace, is to re-open in November with a new name - The Garden Museum.
Dow Jones Architects was chosen to redesign the interior of the historic building following a national competition, and the firm's vision for a series of boxed galleries will become a reality when the Swiss-made wooden boxes arrive next month.
Museum of Garden History director Christopher Woodward said the revamp was part of a drive to look to the future of design, as well as its past.
"It was evident that people wanted to address the contemporary heritage," explained Woodward.
"We are taking out 30 years of partitions and MDF and putting inside the church a series of wooden boxes, which will include a mezzanine gallery to display our permanent collection."
The £500,000 project, funded by donations, will culminate in the museum's re-opening on 18 November with a retrospective of Beth Chatto.
The exhibition will examine Chatto's approach to design through archives, paintings and photographs, and explore influences on her work, including her husband Andrew's study of plants, the work of artist Sir Cedric Morris and her interest in music and architecture.
"We are trying to create a place where people debate contemporary issues and will be doing a retrospective on a different gardener each year," added Woodward. "It is very exciting."