They were created by an idiot. I caught and fixed the problem of the name in the .cue file not matching the actual name of the .ape file but I still can't get anything usable.
Digging around on the web I found the program Burrrn suggested--it's at least willing to try it. It's a beta, though, and that became apparent when part way through the burn it started complaining about various things and it produced a disk that won't play.
Nero complains: "Cannot open the file ", followed by "An error occurred while loading the cue sheet file ". The line is the line that specifies the .ape file. The path is correct, Burrrn found it.
Imgburn complains: "ConnectFilters(Source, Sample Grabber) Failed! Reason: No combination of intermediate filters could be found to make the connection." It at least found the .ape file.
VLC can play the file although I haven't attempted to listen to the whole thing.
How do I stop this file from making a monkey of my efforts???
VLC says: Codec: Monkey's Audio (APE)
I forgot to add: The titles in the .cue file are in some east Asian language, almost certainly either Chinese or Japanese.
@Moab: I simply set out to burn it--it's supposedly derived from a CD, it should be possible to put it back on a CD. It's a series of tracks.