We found this is it mold

IMO you are making several leaps here that don't fit. You've got mold below a window, spread out about 2 feet on either side, and you suspect condensation as the culprit.

You specify that the damage is old - years old by the look of it. Condensation happens every year, every season. Why would the damage be old, it should be fresh if it's condensation.

Finally - if condensation were the culprit, the condensation and thus the mold would be uniform. Water leakage through a window, however, is going to fan out below the window in both directions...

The answer is your window leaked at some point - years ago by the sound of it since the damage is old.

Solution?

Remove and replace ONLY the section of drywall that is damaged. Do not add a partial vapor barrier as that's about as useful as putting in an aquarium with only 2 sides. Mud and tape properly and repaint.

I really don't have time to write a good answer, but I have and others have written some good responses that would answer your question. Look at the tags for mold. ID is important before remediation can be done. You have to know what kind of mold you are dealing with. Some minor mold can be dealt with DIY, but other toxic molds need professional attention. Samples need to be taken and IDed by a lab. Don't take chances, get it right.