The statistical mechanics of polymerized surfaces with a finite bending rigidity is studied via the Monte Carlo method. The model system consists of a hexagon, L atoms across, excised from a triangular lattice embedded in three-dimensional space. Nearest-neighbor atoms interact via an infinite-square-well potential, while the bending energy is proportional to the (negative) scalar product of unit normals to adjacent triangles. Self-avoiding interactions are not included. The largest hexagon considered (L=19) consists of 271 atoms. Unlike linear polymers or liquid membranes, these surfaces undergo a remarkable finite-temperature crumpling transition, with a diverging specific heat. For small =/kBT, the surface is crumpled, and the radius of gyration Rg grows as lnL. For large we find that the surface remains flat, i.e., RgL. Our results demonstrate the presence of a finite-temperature (second-order) crumpling transition, and provide a lower bound on a related transition in real self-avoiding membranes.