Stan,
You are getting every recommendation I would make. The key thing is to kill the mold spores, dead, with whatever product you pick, with ordinary bleach, perhaps cut 1/3 strength as good as any, but the fumes are toxic and you should use an organic vapor mask as well as durable nitrile gloves. If there are spores under existing paint, they will come up through it, many cases.
Once the spores are dead, new paint will help precent a recurrence ... provided there is no food source, per Bob's remarks. Heat and ventilation are more effective than the various dry air chemicals, because their water removal capacities are very limited, and they are soon overtaxed in most situations.
Porous surfaces are the devil to disinfect, with highly porous items such as canvas and other fabrics usually better thrown away in lieu of cleaning, unless cleaning occurs frequently.
Mold is pernicious stuff.