This is probably my absolute favorite short story/novella ever. I have it in the collection of Zelazny works entitled
The Last Defender of Camelot.
Mordel has already given a wonderful summary, I just wanted to add my wholehearted recommendation!
A couple more trivial points: Frost was created by Solcom. During his construction, there was a solar flare; Solcom can't remember what happened (he's orbital, so was affected) but when he came out of it, there was Frost, complete. This is why Frost has a name instead of a designation (like the Beta-machine).
Also, the last refuge of humanity was apparently in a South American town named Bright Defile.