Rear Window Brake Light Not Working on a 328

Need help fixing this one.

Last year I noticed that the brake light in my rear window doesn't come on when I apply the brakes (discovered it in my garage when I didn't see a reflection on the back of the garage door). I replaced the bulb but that didn't help.

When checking it out I noticed two things. First, when I applied the brakes, the light did come on, but only after a couple of seconds and it was extremely dim. I put a volt-meter over the two leads of the lights' connector and noticed that when the brake is not applied, there is continuity between the power and ground. Is that normal?

Any advise you can give would be great.