I recently bought a 1998 Honda Civic DX and discovered that my brake
lights are not working.

I began troubleshooting the problem by first replacing the brake light
switch (under the brake pedal) which didn't help and then replaced the
third brake light bulb (which was burnt out) but that didn't help
either.

I then started looking at the brake light bulbs and discovered
something a little odd.  The actual brake light bulb has two filaments
in it - the first is for the running lights and the second gets
illuminated when the brake pedal is pressed.  What's happening with my
car is that if the headlights are on, both filaments are lit for some
unknown reason.  I should add that the turn signals still work so I
know that there is power in the circuit.

Any ideas or suggestions on where I should look to fix the problem?

Thanks in advance,
Christopher