Resonant structures in B0→ψ′π-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+π- resonances alone, which is confirmed with a model-independent approach. A highly significant Z(4430)-→ψ′π- component is required, thus confirming the existence of this state. The observed evolution of the Z(4430)- amplitude with the ψ′π- 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+.