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The rate of the final step in the astrophysical αp process, the ^{34}Ar(α,p)^{37}K reaction, suffers from large uncertainties due to a lack of experimental data, despite having a considerable impact on the observable light curves of x-ray bursts and the composition of the ashes of hydrogen and helium burning on accreting neutron stars. We present the first direct measurement constraining the ^{34}Ar(α,p)^{37}K reaction cross section, using the Jet Experiments in Nuclear Structure and Astrophysics gas jet target. The combined cross section for the ^{34}Ar,Cl(α,p)^{37}K,Ar reaction is found to agree well with Hauser-Feshbach predictions. The ^{34}Ar(α,2p)^{36}Ar cross section, which can be exclusively attributed to the ^{34}Ar beam component, also agrees to within the typical uncertainties quoted for statistical models. This indicates the applicability of the statistical model for predicting astrophysical (α,p) reaction rates in this part of the αp process, in contrast to earlier findings from indirect reaction studies indicating orders-of-magnitude discrepancies. This removes a significant uncertainty in models of hydrogen and helium burning on accreting neutron stars.