The neutron capture cross section of elemental rhenium was measured to neutron energies between 3 and 1900 keV at the Oak Ridge Electron Linear Accelerator time-of-flight facility. A deformed optical model was used to analyze published neutron total cross sections for rhenium and low-energy average resonance parameters for 185Re and 187Re. The optical model results were used with reaction theory to calculate radiative capture cross sections for comparison with the present experimental data.