Thanks, John. This is the second roll of film I've shot with this camera and the first time I encountered this issue. The first roll turned out perfect or so I thought until I checked the images and saw one that had the same issue.
Between the two rolls, I tried to remove one old stuck screw from the bottom plate and ended up popping the head of it off. So I drilled down that screw hole and fit a slightly larger one in.
My immediate reaction to this black band was the drilling messed something up inside. I'm sort of glad now that this problem existed before opening the bottom plate.
You are right--if I remember correctly, all of the photos with this issue were taken at 1/1000 shutter speed. Do you happen to know off the top of your head how expensive of a job this will be or whether it might be better to invest money in another Nikon FM?