Who would have thought that after four years without a lit odometer, I would be able to get it to work again. The Buick dealer had told me I would have to replace the entire instrument cluster, so it remained unrepaired. However, a recent Google search showed that the problem was a common one and Circuit Solutions even had a YouTube video about how to make the repair to the circuit board in the instrument cluster.
Never would have dreamed that I could get the whole dash out without breakage or loss of parts, but I did. Packed the whole instrument cluster up, shipped it via UPS on a Monday morning and received the repaired unit Saturday afternoon. By Sunday, it was back in the car and all was right in the world. Thumbs up!