At first even this process did not work for me (going to itunes.com/download). I would click on the "Download now" button, it would show me the next screen, but nothing would happen. Then I made sure that iTunes and all its helper processes were killed before doing this. This means, of course, shut down iTunes but also look for (in Task Manager) ituneshelper.exe and ipodhelper.exe (and any other processes that look like they may have to do with itunes). Kill these processes.
After that going to the URL and downloading itunes did work. I was then able to run the installer and get upgraded to the latest version of iTunes.