Partial γ-ray production cross sections and the total radiative thermal-neutron capture cross section for the Re185(n,γ)Re186 reaction were measured using the Prompt Gamma Activation Analysis facility at the Budapest Research Reactor with an enriched Re185 target. The Re186 cross sections were standardized using well-known Cl35(n,γ)Cl36 cross sections from irradiation of a stoichiometric natReCl3 target. The resulting cross sections for transitions feeding the Re186 ground state from low-lying levels below a cutoff energy of Ec=746keV were combined with a modeled probability of ground-state feeding from levels above Ec to arrive at a total cross section of σ0=111(6)b for radiative thermal-neutron capture on Re185. A comparison of modeled discrete-level populations with measured transition intensities led to proposed revisions for seven tentative spin-parity assignments in the adopted level scheme for Re186. Additionally, 102 primary γ rays were measured, including 50 previously unknown. A neutron-separation energy of Sn=6179.59(5)keV was determined from a global least-squares fit of the measured γ-ray energies to the known Re186 decay scheme. The total capture cross section and separation energy results are comparable to earlier measurements of these values.