The cross section of the $^{23}$Na($n, gamma$)$^{24}$Na reaction has been measured via the activation method at the Karlsruhe 3.7 MV Van de Graaff accelerator. NaCl samples were exposed to quasistellar neutron spectra at $kT=5.1$ and 25 keV produced via the $^{18}$O($p, n$)$^{18}$F and $^{7}$Li($p, n$)$^{7}$Be reactions, respectively. The derived capture cross sections $langlesigmarangle_{rm kT=5 keV}=9.1pm0.3$ mb and $langlesigmarangle_{rm kT=25 keV}=2.03 pm 0.05$ mb are significantly lower than reported in literature. These results were used to substantially revise the radiative width of the first $^{23}$Na resonance and to establish an improved set of Maxwellian average cross sections. The implications of the lower capture cross section for current models of $s$-process nucleosynthesis are discussed.
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