Rather than have the cable at it's bitter end after cutting away the frayed section, I decided to replace it. I rode the commuter bike to the local bicycle emporium, and rolled it inside. As it sat there among the shiny new Treks, Electras and Raleighs, I asked if they had the needed cable in stock.
I was surprised to learn they did indeed stock the piece. I had thought it would be an obsolete part that would require ordering. For three bucks, I was back on the road, headed for home with a replacement clipped in the basket. Ten minutes of wrenching, and I had the new cable on the bike, the cut end soldered, and the brakes adjusted. I will have to be more careful next time I lay the bike down.