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Artefacts at the museum have been threatened by inadequate storage conditions, including excessive moisture and poorly controlled extremes of cold and heat.
Some artefacts have been damaged by rust and borer infestation, with excessive heat also harming some wooden objects.
The first big collection of artefacts - involving domestic appliances, such as ovens and washing machines - was moved into the new climate-controlled storage building at the museum complex by museum and contract staff.
The building constitutes the first stage of the museum's $35 million development plan.
Main contractor Lund South Ltd began building the new storage facility last year.
Museum project development manager Bronwyn Simes said progress had since been "pretty straightforward" and the building was now within a few days of completion.
Recent work had included internal wiring, plumbing and installing the air-conditioning system.
Some more wiring work and commissioning of the air-conditioning system were still under way, she said.
The redevelopment second stage is due to start in February, with the whole project expected to be completed in 2012.