The two properties are located above the former Fairways Garage site.
They will be converted into three one-bedroom flats.
The local authority is investing £307,000 in the refurbishment of the building to create affordable rentals in the town and bring the High Street property back into use.
The project includes some demolition works, the creation of a new access stair and entrances and the addition of a new balcony on the south facing side of the building, to give tenants better outdoor space.
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As well as the conversion of the flats, works include some re-pointing, roughcasting checks and repairs; exterior cleaning and paintwork; and some general roof repairs, including repair and replacement work to the existing cast iron gutters and downpipes.
Discussing the development, a council spokesperson said: "Unfortunately to get the necessary work done safely, and at a reasonable cost, the car park will have to be closed for six weeks from Monday 30 September, while the project is completed.
"We apologise for the inconvenience the closure may cause but it will ensure the contractor is able to do the work safely, without any risk to vehicles and residents. Signs to alternative car parks in the town will be in place."
The contractor is RS Hill, Joiners and Building Contractors, Forfar.
Work will begin on site on Thursday 26 September.
(JP/CD)
