  We generalize Cauchy's celebrated theorem on the global rigidity of convex
polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle
polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and
proper non-unitary c-polyhedra with M\"obius-congruent faces that are
consistently oriented are M\"obius-congruent. Our result implies the global
rigidity of convex inversive distance circle packings in the Riemann sphere as
well as that of certain hyperideal hyperbolic polyhedra in $\mathbb{H}^{3}$.
