Robert Conneely has been given the go ahead to renovate the properties to the front and rear of 4 and 5 Market Street.
The development will return the property to two 3-storey residential units.
The residential property at 4 and 5 Market Street previously accommodated a funeral and undertakers business with a workshop at the back.
The works to the property involve the partial removal of the rear element of the building.
It will be rebuilt to form 2 four-bedroom houses.
In granting permission, city planners imposed 11 conditions.
One condition states that a suitably qualified archaeologist must be hired to carry out pre-development testing on the site.
Also, the developer must provide details of the new door mechanism to No.6 Market Street next door to allow for secure entry and emergency exit.