  In this paper we perform a fine blow-up analysis for a fourth order elliptic
equation involving critical Sobolev exponent, related to the prescription of
some conformal invariant on the standard sphere. We derive from this analysis
some a priori estimates in dimension 5 and 6. On the five dimensionl sphere
these a priori estimates combined with the perturbation result of Part I allow
us to obtain some existence results. On the six dimensional sphere we prove the
existence of at least one solution when an associated index is different from
zero.
