Wim Decorte answered this question in another thread: he suggested that the files be in the same folder on the server (should be on the same server, of course).
That did the trick - after a slight adjustment to the EDS definition.
* The file was moved so they were both in the same folder
* I edited the EDS from "fmnet:/fms.server.com/ExtFile" to "file:ExtFile"
(I made the edits via the 'Edit -> Add File -> Remote ->...' process; I wasn't just manually editing the lines.)
Then my error went away! Thanks Wim.