  Let B be a sigma-unital C*-algebra. We show that every strongly continuous
E_0-semigroup on the algebra of adjointable operators on a full Hilbert
B-module E gives rise to a full continuous product system of correspondences
over B. We show that every full continuous product system of correspondences
over B arises in that way. If the product system is countably generated, then E
can be chosen countable generated, and if E is countably generated, then so is
the product system. We show that under these countability hypotheses there is a
one-to-one correspondence between E_0-semigroup up to stable cocycle conjugacy
and continuous product systems up isomorphism. This generalizes the results for
unital B to the sigma-unital case.
