Resonant structures in B-0 -> psi''pi K--(+) decays are analyzed by performing a four-dimensional fit of the decay amplitude, using pp collision data corresponding to 3 fb(-1) collected with the LHCb detector. The data cannot be described with K+pi(-) resonances alone, which is confirmed with a model-independent approach. A highly significant Z(4430)(-) -> psi''pi(-) component is required, thus confirming the existence of this state. The observed evolution of the Z(4430)(-) amplitude with the psi''pi(-) mass establishes the resonant nature of this particle. The mass and width measurements are substantially improved. The spin parity is determined unambiguously to be 1(+).