Step 1: Disassembly
Once the halves were separated (photo 1), I carefully pried the clock hands free (photo 2), and un-clipped the remains of the neon light tube (which was broken, of course).
Next, I flipped the base over and removed the old clock motor and gear assembly and the transformer for the neon light (photo 3.
Step 2: Identifying the problem(s)
The clock motor could be replaced with a quartz motor, but replacing the neon tube would cost much more than the clock was worth. So, I took a slightly different approach in replacing it.