Are you doing a Software Update or downloading directly from Apple?
If Software Update, instead download the combo 10.6.8 update from here. Use Disk Utility (on your hard drive in application/utilities) to repair permissions, then apply the combo update, and when done, repair permissions again. You'll be at 10.6.8 at that point.
Thank you for responding.
I repaired my permissions and when I tried to download the update you linked to, it failed twice with a "mounting failed" message. When I double click the message, it says I can't open the file because it is still being downloaded (but it is not).
I greatly appreciate your help!