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This work presents a direct measurement of the $^{96}$Ru($p, \gamma$)$^{97}$Rh cross section via a novel technique using a storage ring, which opens opportunities for reaction measurements on unstable nuclei. A proof-of-principle experiment was performed at the storage ring ESR at GSI in Darmstadt, where circulating $^{96}$Ru ions interacted repeatedly with a hydrogen target. The $^{96}$Ru($p, \gamma$)$^{97}$Rh cross section between 9 and 11 MeV has been determined using two independent normalization methods. As key ingredients in Hauser-Feshbach calculations, the $\gamma$-ray strength function as well as the level density model can be pinned down with the measured ($p, \gamma$) cross section. Furthermore, the proton optical potential can be optimized after the uncertainties from the $\gamma$-ray strength function and the level density have been removed. As a result, a constrained $^{96}$Ru($p, \gamma$)$^{97}$Rh reaction rate over a wide temperature range is recommended for $p$-process network calculations.