  We calculate the total derived functor for the map from the Weil-etale site
introduced by Lichtenbaum to the etale site for varieties over finite fields.
In particular, there is a long exact sequence relating Weil-etale cohomology
and etale cohomology. In the second half of the paper, we apply this to study
the Weil-etale cohomology of the motivic complex for smooth and projective
varieties. These groups are expected to be finitely generated, to give an
integral model for l-adic cohomology, and to be related to special values of
the zeta function. We give necessary and sufficient conditions for this to
hold, and examples.
