An amplitude analysis of an exclusive sample of 5765 events from the reaction pi- p-->eta'pi- p at 18 GeV/c is described. The eta'pi- 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 eta'pi-. The dominant, exotic (non- qq) 1(-+) partial wave is shown to be resonant with a mass of 1.597+/-0.010(+0.045)(-0.010) GeV/c2 and a width of 0.340+/-0.040+/-0.050 GeV/c2. This exotic state, the pi1(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.