An amplitude analysis of an exclusive sample of 5765 events from the reaction π − p → η ′ π − p at 18 GeV/c is described. The η ′ π − production is dominated by natural parity exchange and by three partial waves: those with J P C = 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 ′ π − . The dominant, exotic (non-qq) 1 −+ partial wave is shown to be resonant with a mass of 1.597 ± 0.0102 and a width of 0.340 ± 0.040 ± 0.050 GeV/c 2 . 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.