The efficiency of the weak s process in low-metallicity rotating massive stars depends strongly on the rates of the competing ^{17}O(α,n)^{20}Ne and ^{17}O(α,γ)^{21}Ne reactions that determine the potency of the ^{16}O neutron poison. Their reaction rates are poorly known in the astrophysical energy range of interest for core helium burning in massive stars because of the lack of spectroscopic information (partial widths, spin parities) for the relevant states in the compound nucleus ^{21}Ne. In this Letter, we report on the first experimental determination of the α-particle spectroscopic factors and partial widths of these states using the ^{17}O(^{7}Li,t)^{21}Ne α-transfer reaction. With these the ^{17}O(α,n)^{20}Ne and ^{17}O(α,γ)^{21}Ne reaction rates were evaluated with uncertainties reduced by a factor more than 3 with respect to previous evaluations and the present ^{17}O(α,n)^{20}Ne reaction rate is more than 20 times larger. The present (α,n)/(α,γ) rate ratio favors neutron recycling and suggests an enhancement of the weak s process in the Zr-Nd region by more than 1.5 dex in metal-poor rotating massive stars.