I just use bleach straight out of the jug to kill it, but most would recommend diluting the bleach 10:1 with water to make a safer solution to use when wearing clothing that could be bleached as a result.
Truth is that ordinary black bathroom mold is not at all hard to kill, and even a not-so-mild solution of TSP in water will kill it as dead as bleach will. TSP has the added advantage that it also dulls the gloss of drying oil based paints, so the reason why they used to recommend cleaning walls with TSP prior to painting is that when walls still had linseed oil based paint on them, the TSP would etch the surface of the paint, giving that surface "tooth" so that the next coat would adhere better. So far as I know, TSP doesn't dull the gloss of alkyd or emulsion paints.
What I would do is clean that ceiling (and prolly the walls, too) with bleach and then rinse the bleach off with clean water and allow the walls to dry. Then, if the paint on the walls is glossy, give it a light sanding to dull that gloss so the next coat sticks better. Alternatively, use a chemical deglosser like Liquid Sandpaper (which I have never used).
Then, paint over the walls or ceiling with a PAINT SPECIFICALLY MADE FOR BATHROOMS, like Zinsser's PermaWhite Bathroom paint. The reason why is that paints specifically made for bathrooms will:
a) be made out of Perspex plastic resins that are by their very nature much more immune to softening up and/or losing their adhesion under humid conditions, but anyone making a paint for bathrooms will have gone to the trouble of selecting the particular Perspex resin that is MOST immune to this, and...
b) will have a mildewcide added to the paint that will migrate to the surface of the paint (as a result of it's affinity for water) to kill any mildew spores that grow on the paint before they have a chance to grow, thereby keeping the paint free of mildew in future.
TWO PS's to add:
PS#1:
Contrary to popular belief, painting a bathroom wall with an OIL BASED paint is shooting yourself in the foot. Oil based paints are made from drying oils or the fatty acids from drying oils, and mildew will use those as a food source. So, painting your walls with a linseed oil based paint is the rough equivalent of painting it with food for mildew. Alkyd paints are less susceptible to attack by mildew, but acrylic paints are the least susceptible cuz mildew can't eat Perspex.
PS#2:
The mildewcides that are added to bathroom paints gradually leach out of the paint, and when they do the paint film is no longer protected from mildew growing on it. Here, go to this web page:
http://www.pcimag.com/CDA/Archives?issue=298
and download the article entitled:
Chemistry is Key to Microbiology for Coatings - Posted: 6/1/2000
And look at Figures 5 and 6 of that article.
Figure 5 shows the rate at which different mildewcides leach out of a semi-gloss paint film. Obviously, some leach out much more rapidly than others. One of which (the very bottom line that hugs the horizontal axis, barely leaches out at all).
Figure 6 shows the relative rates at which a particular mildewcide will leach out of 5 flat paints, three semi-gloss paints and one gloss paint. In the case of the 5 different flat paints, MOST of the mildewcide comes out of the paint rapidly, and the result is that the paint has way more protection against mildew than it needs for a short time, and then becomes susceptible to mildew once the reserve of mildew within the paint is depleted. In the case of the gloss paint, the mildewcide doesn't come out of the paint fast enough to be effective, so that paint may be susceptible to mildew right from the start. The trick is to match the TYPE of mildewcide used in the paint with the gloss level of the paint to have the mildewcide come out at the MINIMUM rate that is still effective in killing all of the mildew spores that land on the paint.
This is why if someone just tosses a packet of mildewcide in a can of paint before shaking the paint, the results are completely unpredictable because no testing has been done to find out how quickly that mildewcide will leach out of that paint. (But, I'd do that too if I was only concerned about the paint lasting 1 year until my warrenty expired.) You're much better off buying a paint where the manufacturer has done the tweaking to match the type of mildewcide in the paint with the gloss level of the paint to achieve the slowest leaching that is still effective, and therefore the longest possible protection against mildew.
