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The new development is on target to be completed by October and garden team leader Alan Matchett said all attention would then turn to the big move.
He said the new facilities would be thoroughly tested before being entrusted with the many plants housed by the old buildings across Lovelock Ave - a biological library of specimens from around the world.
Mr Matchett said the shift could take two months and might not be complete until after Christmas.
''It will be a big job ... we will test all equipment to make sure it is working because we have got to get it right.''
Once the move was complete the old glasshouses would be demolished and they could start looking at the next stage of development, including more gardens, public toilets, an information area and lecture theatre.
Mr Matchett said the project had been in the planning stage since 2006, so it was great to get the first part under way.
The new development made much better use of space, should be more energy-efficient and was better designed to cater for larger trees and different climates.
It would also be able to host tour groups and groups of pupils.
''We will have a facility now that we can actually bring people through with no health and safety issues and we hope to do more teaching.''
The new development also has large rain-collection tanks to offset water usage and a modern wood chip boiler.
Open days would be held in the New Year for members of the public to see what had been built, he said.