Hi, Gary. The process that you will have to use for getting rid of the black stain caused by the water takes a while but is pretty easy and uses pretty common products. First strip the finish off the affected area. You have to do this since some of the black is probably under the finish. You just have to do the affected boards. I prefer a relatively safe stripper like Citristrip. Just follow label directions. Rinse the area with mineral spirits/paint thinner after you have stripped. Let this dry for a day and take the rag outside for a couple of days to dry completely.
Then wipe the area with household bleach to kill any living mildew. Follow with a wipe down with white vinegar to neutralize the bleach and let this dry for a day or so.
Now you are finally ready to go after the black spot. Oxalic acid is sold under the trade name Wood Bleach at the home centers. Use this following label directions. When it has worked, neutralize with a cup of bleach to a gallon of water and let the wood dry. You are now ready to refinish. You might want to consider using spar varnish as you will want good UV resistance as well as water resistance on the window sill.
Be sure that you use chemical resistant gloves while using any of the chemicals mentioned. Also, wear a dust mask when you are mixing the oxalic acid because the dust is caustic. Your nose will appreciate it.