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The original slates had deteriorated and are being replaced with new ones from the Iberian Peninsula which match the original colours and shapes.
Mr Meegan is a specialist slate craftsman who has been laying slate for nearly 50 years.
The roof timber on the stables has also been repaired and the exterior of the building will be repainted.
The stables had a category 1 historic listing with Heritage NZ, which was looking into the origin of the original slates, a Dunedin City Council spokeswoman said.
The original slates might have come from an old quarry near Timaru.
The work was estimated to cost about $100,000 and was expected to be completed in about a month, depending on the weather.