The former Scotland and Liverpool midfielder, and boyhood Dundee fan, was a free agent and both club and player have never hidden their desire to make the move happen.
The 34-year-old, who had been training with Bolton Wanderers, has pulled on the dark blue shirt in two testimonials but will soon be doing it for real.
In a statement announcing the transfer coup, Dundee said: “Negotiations surrounding the deal have gone on for a number of weeks but we are delighted to agree a deal that suits all parties.
“For the financial stability of the club we had to stay out with the financial plan created to help us through the 2020/21 season.
“We were open to the fact that this meant we would have to get creative to make this deal happen and we would like to place on record our thanks to three current club sponsors, without whom it would not have been possible.”
Adam will wear the number 26 for his new team in the Championship this season.
