Recently, the triple-α (3α) process, by which three 4 He nuclei are fused into a 12 C nucleus in stars, was studied by different methods in solving the quantum mechanical three-body problem. Their results of the thermonuclear reaction rate for the process differ by several orders at low stellar temperatures of 10 7 − 10 8 K. In this paper, we will present calculations of the 3α process by a modified Faddeev three-body formalism in which the long-range effects of Coulomb interactions are accommodated. The reaction rate of the process is calculated via an inverse process, three-alpha (3α) photodisintegration of a 12 C nucleus. Calculated reaction rate is about 10 times larger than that of the Nuclear Astrophysics Compilation of Reaction Rates (NACRE) at 10 7 K, and is remarkably smaller than the results of the recent three-body calculations. We will discuss a possible reason of the difference.