[en] An amplitude analysis of an exclusive sample of 5765 events from the reaction π^- p → η(prime)π^- p at 18 GeV/c is described. The η(prime)π^- production is dominated by natural parity exchange and by three partial waves: those with J^PC = 1^-+, 2⁺⁺, and 4⁺⁺. A mass-dependent analysis of the partial-wave amplitudes indicates the production of the a₂(1320) meson as well as the a₄(2040) meson, observed for the first time decaying to η(prime)π^-. The dominant, exotic (non-q(bar q)) 1^-+ partial wave is shown to be resonant with a mass of 1.597 ± 0.010_-0.010^+0.045 GeV/c² and a width of 0.340 ± 0.040 ± 0.050 GeV/c². This exotic state, the π₁(1600), is produced with a t dependence which is different from that of the a₂(1320) meson, indicating differences between the production mechanisms for the two states