The efficiency of the slow neutron-capture process in massive stars is strongly influenced by neutron-capture reactions on light elements. At low metallicity, 16 O is an important neutron absorber, but the effectiveness of 16 O as a light-element neutron poison is modified by competition between subsequent 17 O(α, n) 20 Ne and 17 O(α, γ ) 21 Ne reactions. The strengths of key 17 O(α, γ ) 21 Ne resonances within the Gamow window for core helium burning in massive stars are not well constrained by experiment. This work presents more precise measurements of resonances in the energy range E c.m. = 612-1319 keV. We extract resonance strengths of ωγ 638 = 4.85 ± 0.79 μeV, ωγ 721 = 13.1 +3.2 −2.4 μeV, ωγ 814 = 7.72 ± 0.55 meV, and ωγ 1318 = 136 ± 13 meV, for resonances at E c.m. = 638, 721, 814, and 1318 keV, respectively. We also report an upper limit for the 612 keV resonance of ωγ < 140 neV (95% c.l.), which effectively rules out any significant contribution from this resonance to the reaction rate. From this work, a new 17 O(α, γ ) 21 Ne thermonuclear reaction rate is calculated and compared to the literature. The effect of present uncertainties in the 17 O(α, γ ) 21 Ne reaction rate on weak s-process yields are then explored using postprocessing calculations based on a rotating 20M low-metallicity massive star. The resulting 17 O(α, γ ) 21 Ne reaction rate is lower with respect to the preexisting literature and found to enhance weak s-process yields in rotating massive star models.