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

The decomposition of global conformal invariants II: The Fefferman-Graham ambient metric and the nature of the decomposition