try this

Here is a link that might be useful: mold killer

A similar but cheaper solution can be made from:
1 quart hot water
1 tablespoon baking soda (sodium bicarbonate)
2 tablespoons washing soda (sodium carbonate) aka soda ash
2 tablespoons trisodium phosphate (TSP)

Thanks, christophern and Renovator8! I appreciate the tips. This will be my weekend project.

My impatience got the best of me and I went after a little spot with undiluted bleach on a q-tip. It seemed to clear it up. I don't know about the long term effect of bleach on wood, so this probably isn't a good idea. I will try a less toxic method.

I can't seem to shake sinus problems after a bout with the flu three weeks ago. I suspect the mold is exacerbating the condition. I will be wearing gloves, a mask and goggles when I tackle the mold. I hope it hasn't spread all over the house.

I will feel less anxious when the mold is gone, floors and walls are repaired and covered with drywall and floor. (And the exterior rat access spots are covered.) Oh yeah, and the old termite damage is repaired!

It will be worth it in the end!

I found your original description confusing but I assume the mold is on the inside of the stud cavity which is now exposed from the inside of the house. The most important issue is how to prevent water from entering again. If that is not addressed, there is no point in doing much in the cavity because the mold will return. The surface of the stucco is not the weather barrier. There should be a barrier behind it that drains. If that is damaged, the surface repair will be temporary.

If you are going to go near the mold at least get a mask rated at N100 but disturbing the mold with bleach may be what is complicating your health. Undiluted bleach is very toxic especially in sunlight. Even dead mold spores can cause allergic reactions and should not be allowed to get into the air.

You should not be trying to "kill" the mold but to remove it or any damaged soft material without spreading the dead or living spores. This is not a casual DIY project. Either find someone who knows how to do it or do enough research to be able do it correctly on your own. Right now you should not be touchng the walls.

Here is a link that might be useful: bleach

Vinegar (especially concentrated 20-50% acidity ) was the best solution for eliminate any mold. I used this method for whole my life. Just try and you will be surprised! No toxic chemicals only organic!

Here is a link that might be useful: Eliminate mold with vinegar

Thanks for all of the suggestions about removing the mold, and particularly the messages about finding and fixing the source.

Here is an update...
When I went to remove the mold wearing all sorts of protective gear and armed with lots of supplies (toxic and non), I realized that the wetness emanated from behind the studs which covered the corner AND that there was a lot of dampness hidden in the corner.

You can predict the rest of the story...

The studs came out when the contractor arrived and so much more mold and rotten wood was exposed. Clearly, there was a baaaad leak on the exterior wall that had been leaking for a long time and had rotted out the wood in the sill and plate. Forget wiping off the mold, it was time to fix the leak and remove all of the soggy rotten moldy wood. The tar paper between the layers of stucco and wood framing had disintegrated in the wet areas. After two weeks, lots of stucco crack sealing, some new flashing, rain and hose spraying, there are no more leaks in this area. The wet wood is gone.

We will wait to close up the area until we are sure that there are no more leaks.

Thanks again for the suggestions and moral support!