The cross section of the S(n, n) Si reaction was measured in a neutron energy spectrum yielding the stellar rate for a thermal energy of kT=25 keV. For the Cl(n, p) S reaction, the experiment was performed in the neutron energy range from 10 to 260 keV. Our new result for S resolves the problem of a large discrepancy between two previous experiments. In case of Cl it could be shown that the (n, cx) channel is negligibly small compared to the (n, p) reaction at s-process conditions. The present results are used for a detailed discussion of a possible s-process origin of the rare neutron-rich isotope S. PACS number(s): 25.40.Ny, 97.10.Cv, 27.30.+t, 29.40.Cs