Thank you. The final snippet of .htaccess code that works for me turns out to be:

Action my-handler /cgi-bin/index.cgi
SetHandler my-handler
[download]

And the requested path is available to the script in the PATH_INFO environment variable.

UPDATE: For this to work, a file named index.cgi must be available in the same directory as the .htaccess file. The file may be empty or contain anything, since it is never executed. (Question: why is this file needed by Apache?)