The reaction rate of the triple-alpha (3α) process at low temperatures, where resonant reaction is not dominant, is calculated through the inverse process, the photodisintegration of a 12C nucleus. For this, Schrödinger equations in a three-alpha (3-α) model of 12C are directly solved by a Faddeev method, which has been successfully applied to three-nucleon problem so far. The nuclear Hamiltonian consists of an α-α potential, which reproduces the 8Be resonance state, together with three-body potentials to reproduce 12C properties. Our results of the 3α reaction rate are about 103 times larger at low temperature (T = 107 K) than a standard rate from the Nuclear Astrophysics Compilation of Reaction Rates (NACRE), which means our results are remarkably smaller than recent results of quantum-mechanical three-body calculations by Ogata et al.