  Let \Sigma_g be a closed orientable surface of genus g \geq 2 and \tau a
graph on \Sigma_g with one vertex which lifts to a triangulation of the
universal cover. We have shown that the cross ratio parameter space
\mathcal{C}_\tau associated with \tau, which can be identified with the set of
all pairs of a projective structure and a circle packing on it with nerve
isotopic to \tau, is homeomorphic to \mathbb{R}^{6g-6}, and moreover that the
forgetting map of \mathcal{C}_\tau to the space of projective structures is
injective. In this paper, we show that the composition of the forgetting map
with the uniformization from \mathcal{C}_\tau to the Teichm\"uller space
\mathcal{T}_g is proper.
