  This is the second in a series of papers where we prove a conjecture of Deser
and Schwimmer regarding the algebraic structure of ``global conformal
invariants''; these are defined to be conformally invariant integrals of
geometric scalars. The conjecture asserts that the integrand of any such
integral can be expressed as a linear combination of a local conformal
invariant, a divergence and of the Chern-Gauss-Bonnet integrand.
  The present paper addresses the hardest challenge in this series: It shows
how to {\it separate} the local conformal invariant from the divergence term in
the integrand; we make full use of the Fefferman-Graham ambient metric to
construct the necessary local conformal invariants, as well as all the author's
prior work [1,2,3] to construct the necessary divergences. This result combined
with [3] completes the proof of the conjecture, subject to establishing a
purely algebraic result which is proven in [6,7,8].
