Thank you. The final snippet of .htaccess code that works for me turns out to be:
in reply to Re: Ways for capturing all requests under Apache
in thread Ways for capturing all requests under Apache
Action my-handler /cgi-bin/index.cgi
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?)