  The competing $^{22}$Ne($\alpha,\gamma$)$^{26}$Mg and
$^{22}$Ne($\alpha,n$)$^{25}$Mg reactions control the production of neutrons for
the weak $s$-process in massive and AGB stars. In both systems, the ratio
between the corresponding reaction rates strongly impacts the total neutron
budget and strongly influences the final nucleosynthesis. The
$^{22}$Ne($\alpha,\gamma$)$^{26}$Mg and $^{22}$Ne($\alpha,n$)$^{25}$Mg reaction
rates was re-evaluated by using newly available information on $^{26}$Mg given
by various recent experimental studies. Evaluations of The evaluated
$^{22}$Ne($\alpha,\gamma$)$^{26}$Mg reaction rate remains substantially similar
to that of Longland {\it et al.} but, including recent results from Texas A\&M,
the $^{22}$Ne($\alpha,n$)$^{25}$Mg reaction rate is lower at a range of
astrophysically important temperatures. Stellar models computed with NEWTON and
MESA predict decreased production of the weak branch $s$-process due to the
decreased efficiency of $^{22}$Ne as a neutron source. Using the new reaction
rates in the MESA model results in $^{96}$Zr/$^{94}$Zr and
$^{135}$Ba/$^{136}$Ba ratios in much better agreement with the measured ratios
from presolar SiC grains.
