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A decoration is a choice of circle about each vertex of the surface. Our decorated surfaces are closely related to inversive distance circle packings, canonical tessellations of hyperbolic surfaces, and hyperbolic polyhedra. Discrete conformal equivalence introduces an equivalence relation on decorated piecewise hyperbolic and spherical surfaces. We prove the corresponding uniformization theorem\u2014in each equivalence class there is a unique representative on a constant curvature surface. Furthermore, we introduce a fundamental invariant characterizing the discrete conformal classes. We show that one can deform continuously between decorated piecewise hyperbolic, Euclidean, and spherical surfaces sharing the same invariant. 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