Eventually the only solution I found was to install Acrobat 8.0 on Windows XP instead of Vista.
Adobe gave me one more piece of advice which (while reasonable) did not fix the problem I had. I think it's worth a try at least.
Also, In the Adobe forum related to this is another 'fix' proposed by a user.
Adobe Forums - Error 1935 - An error occured during the installation assembly
Adobe recommended the following:
"We recently have found a solution that has worked a many machines to
resolve the error #1935.#
1. Click #Start#, in Search Box type "Command Prompt" and enter (don't
2. Right click and select "Run as Administrator"
3. Then type and enter:
fsutil resource setautoreset true c:\
4. A message stating, "operation completed successfully# should appear.
Then go ahead and restart your computer.
After the restart is complete, try to install Adobe Acrobat.