There's a TSB on this problem, call a dealership because it might be a free repair via recall.  
I think it's this one:
2006 Chevrolet Cobalt
Bulletin #: 3421B
Sequence #: 10014807
Date: 01/01/1901
Component Name: 110000 electrical system
Make: Chevrolet
Model: Cobalt
Year: 2006
Summary: Engine stalls, loss of electrical systems, and no diagnostic trouble code (dtc). including 2005-06 pontiac pursuit. *tt *sc