This could be a shot in the dark, but hey ... you said that the new alternator lasted one week ... is the battery OK? A toasted battery (one or more cells internally shorted) can drain too high a current from the alternator, causing its premature failure.

This doesn't explain the stalling during idling, however, a recent and painful experience taught me that a misfire can be caused by crappy ignition cables ... my car's engine also died on me a few times during idling because the idle was really rough ...so maybe you have two overlapping problems here. Let us know ...