Lighter heavy elements beyond iron and up to around silver can form in neutrino-driven ejecta in core-collapse supernovae and neutron star mergers. Slightly neutron-rich conditions favour a weak r-process that follows a path close to stability. Therefore, the beta decays are slow compared to the expansion time scales, and ($alpha$,n) reactions become critical to move matter towards heavier nuclei. The rates of these reactions are calculated with the statistical model and their main uncertainty, at energies relevant for the weak r-process, is the $alpha$+nucleus optical potential. There are several sets of parameters to calculate the $alpha$+nucleus optical potential leading to large deviations for the reaction rates, exceeding even one order of magnitude. Recently the $^{96}$Zr($alpha$,n)$^{99}$Mo reaction has been identified as a key reaction that impacts the production of elements from Ru to Cd. Here, we present the first cross section measurement of this reaction at energies (6.22 MeV $leq$ E$_mathrm{c.m.}$ $leq$ 12.47 MeV) relevant for the weak r-process. The new data provide a stringent test of various model predictions which is necessary to improve the precision of the weak r-process network calculations. The strongly reduced reaction rate uncertainty leads to very well-constrained nucleosynthesis yields for $Z = 44 - 48$ isotopes under different neutrino-driven wind conditions.
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