A distinct peak is observed in the pi +/- psi' invariant mass distribution near 4.43 GeV in B-->K pi +/- psi' decays. A fit using a Breit-Wigner resonance shape yields a peak mass and width of M=4433+/-4(stat)+/-2(syst) MeV and Gamma=45-13+18(stat)-13+30(syst) MeV. The product branching fraction is determined to be B(B 0-->K -/+Z+/-(4430)) x B(Z+/-(4430)-->pi+/-psi')=(4.1+/-1.0(stat)+/-1.4(syst)) x 10(-5), where Z+/-(4430) is used to denote the observed structure. The statistical significance of the observed peak is 6.5 sigma. These results are obtained from a 605 fb(-1) data sample that contains 657 x 10(6) BB pairs collected near the Upsilon(4S) resonance with the Belle detector at the KEKB asymmetric energy e+ e- collider.