  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.
