A novel technique has been developed, which will open exciting new opportunities for studying the very neutron-rich nuclei involved in the r process. As a proof of principle, the γ spectra from the β decay of ^{76}Ga have been measured with the SuN detector at the National Superconducting Cyclotron Laboratory. The nuclear level density and γ-ray strength function are extracted and used as input to Hauser-Feshbach calculations. The present technique is shown to strongly constrain the ^{75}Ge(n,γ)^{76}Ge cross section and reaction rate.