The cross section of the 23Na(n,γ)24Na reaction was 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 18O(p,n)18F and 7Li(p,n)7Be reactions, respectively. The derived capture cross sections ⟨σ⟩kT=5keV=9.1±0.3 mb and ⟨σ⟩kT=25keV=2.03±0.05 mb are significantly lower than reported in literature. These results were used to substantially revise the radiative width of the first 23Na 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.