An isotope of the 113th element, i.e., 278113, was produced in a nuclear reaction with a 70Zn beam on a 209Bi target. We observed six consecutive α-decays following the implantation of a heavy particle in nearly the same position in the semiconductor detector under an extremely low background condition. The fifth and sixth decays are fully consistent with the sequential decays of 262Db and 258Lr in both decay energies and decay times. This indicates that the present decay chain consisted of 278113, 274Rg (Z=111), 270Mt (Z=109), 266Bh (Z=107), 262Db (Z=105), and 258Lr (Z=103) with firm connections. This result, together with previously reported results from 2004 and 2007, conclusively leads to the unambiguous production and identification of the isotope 278113 of the 113th element.