I think Parallels Desktop ver. 7 has caused this problem for me on my Mac Pro.
I have a Macbook with the same MAC OSX 10.6.8 with the same version of the Mail.app and it works fine.
Mail.App is the default email reader on my MacBook and it can load the .ICS files from invites properly into ICal.
In fact, there are only two choices for default readers.
The Macbook has Parallels Desktop ver. 6 loaded instead of ver. 7.
However, I just remembered that I have a Mac Mini with the same OSX versions with Parallels Desktop ver. 7 installed. It's defaullt email reader is properly set to the Mail.App even though there are many other choices provided by Parallels Desktop.
This leads me to believe that Mr. Glenn Leblanc is correct in that the Mail.App preference settings are corrupt and need to be rebuilt.