The source of the leak has been fixed, however the old mold keeps showing though the new paint.
Kilz is not anti-mold paint.
Once the source of the moisture has been corrected, go to your local Home Depot and buy Concrobium fungicide spray. It's all natural and an encapsulant. It comes in a spray bottle. Follow the directions. This will kill the mold as well as prevent new mold from growing.
Once that is done you can cover the area with Zinnser Bulls Eye 1-2-3 Waterbase Primer. The bulls eye IS mold and mildew resistant. If you have a lot of water stains you may want to go with something stronger like Zinnser Guardz, but Kilz will not help you out in the water stains or mold issue.
It's best to go with a natural, organic and effective remedy. I've been in the business 12 years. First use a roller, dip it in vinegar, ( rice wine or apple cider) Then you have to use a special Lime Paint over a quartz-stone primer. this will eliminate all mold permanently. It raises the alkalinity levels in the air so mold can't grow on it.
You need to kill the mould spores first. First use a sugar soap to thoroughly clean the surface. Then obtain a mould killing chemical (there are lots of them, go to your local hardware store and ask).
Finally, a good sand down and apply a stain blocking primer/undercoat such as those from zinsser or dulux. Ideally something oil based. Then a couple of top-coats of your normal paint and you should be sorted. You can get paint additives or paints that come with anti-mould compounds in them these days. I'd recommend that particularly if it's an area that gets damp often (like a bathroom or laundary).