[en] The activation method is used to measure the stellar neutron capture rates of Re-185 and Re-187 at kT = 25 keV as 1677 + or - 65 and 1269 + or - 62 mbarn, respectively. The experimental setup and procedure are described, and the results are presented in tables and graphs. Also reported are calculations of the capture rates for the unstable isotopes, e.g., 597 + or - 85 mbarn for W-185 and 1775 + or - 265 mbarn for Re-186, again at kT = 25 keV. These results are then applied, using the classical method, to estimate the mean neutron density for the s-process nucleosynthesis branchings at W-185 and Re-186: n(n) = (4.1 + 1.2 or - 1.1) x 10 to the 8th/cu cm. 64 refs