The latest setback was blamed on problems with concrete panels on a new bridge over the Rover Don.
There is also a dispute between the contractors Aberdeen Roads Limited (ARL) and the Scottish Government's Transport Scotland agency about opening the southern section of the road.
The Â£745 million project was originally due to have been finished last winter, which was put back several times, latterly to "late autumn".
Northern sections of the 36-mile road have been opened, between Craibstone and Dyce, Blackdog and Parkhill, and Balmedie and Tipperty.
Mr Matheson said: "Last Friday, Transport Scotland was informed a greater scope of work would be necessary to repair the [bridge] defects.
"They are working hard to repair the defects, and on Monday, the contractor reported it was targeting a December opening date for the whole road."
However, the minister added that ARL "were unable to provide a definitive date for the opening of this section as there are a number of factors which could influence this date including technical issues and other physical factors, such as the weather".
Mr Matheson said: "We will continue to work closely with ARL to ensure everything that can reasonably be done, is being done, and we will provide a definitive date for the opening of this section as soon as possible."
North East Labour MSP Lewis Macdonald said: “It is clear from the transport secretary’s statement that the 20 miles of the AWPR between Craibstone and Stonehaven could be open on Monday morning.
"The only reason that stretch of road remains closed is a contractual dispute between the Scottish Government and Aberdeen Roads Limited.
“Hundreds of millions of pounds of public money have been spent on this project, and it is unacceptable that the public are being prevented from using such a large part of this road simply because of a badly-drafted contract."