  Resonant structures in $B^0\to\psi'\pi^-K^+$ decays are analyzed by
performing a four-dimensional fit of the decay amplitude, using $pp$ collision
data corresponding to $\rm 3 fb^{-1}$ 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
$Z(4430)^-\to\psi'\pi^-$ 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 $1^+$.
