The neutron capture cross section of 185Re was measured in the astrophysically important energy region. Measurements were made using a neutron beam from a 7Li(p,n)7Be neutron source with energies ranging from 3 to 90 keV. Two different experimental techniques, time-of-flight (TOF) and activation methods, were employed. In the TOF experiments, the total neutron capture cross section of 185Re was determined by the pulse-height weighting technique. In the activation method, the partial capture cross section leading to the ground state of 186Re was measured by detecting decay γ-rays from neutron activated samples. The present cross section values were compared with evaluated cross section data and previous measurements. The difference between the TOF and activation results was smaller than experimental uncertainties. This suggests that the production cross section of isomer states of 186Re is very small.