Doggone it! Now it makes sense. The domain account is working. No problem. The configuration of Exchange access was not correctly done. Go back to the machine on which email access work and undo the Exchange email setup. Then, redo it--but pay careful attention to the steps. The original setup stored the settings in the local profile. That is why the user cannot access email from any other location.

I can't remember the specific steps, sorry, but there is a point at which the quicker-clicker will miss a significant checkbox or something to cause this problem.