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 \(\langle\sigma\rangle_{\rm kT=5 keV}=9.1\pm0.3\) mb and \(\langle\sigma\rangle_{\rm kT=25 keV}=2.03 \pm 0.05\) 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.