When you get the "application quit" message, there should be buttons on the window and one of them should be labeled "Report". If you click on that button, you'll get a bunch of technical info about the crash. It will probably contain a clue as to what's going wrong. If you can't get a log this way, go to Applications>Utilities and run Console. Under LOG FILES | ~/Library/Logs | Crash Reporter you should see a file named Mail.crash.log. Send either the log generated when you click the Report button or the log file Mail.crash.log to me in a PM to keep from cluttering up this thread, and I'll take a look at it.
There's probably something corrupted somewhere. Download and run
Preferential Treatment (freeware) and remove any corrupted preference files (there are other preference files associated with Mail). Also download and run
Onyx (also free), repair permissions, and run the other maintenance tasks in Onyx.
Just a hunch... do you know if Growl is installed (it's a helper app, and I've had bizarre things happen in the past due to out-of-date copies of Growl)? Go to System Preferences, then Accounts, and click on the Login Items tab for your account. See if there's an app named GrowlHelperApp listed. If so, you have Growl installed; you can download and install the latest version
here but DON'T DO THIS unless you're sure Growl is already installed.
Finally, are you within reasonable driving distance of an Apple Store? If so, I'd make an appointment at the Genius Bar to take your Mac in and have them look at it. There may also be an authorized Apple dealer in your area that can help you with your problem. See your warranty/Apple Care packet for information on finding an authorized dealer in your area if you're not near an Apple Store.