The O-18(n,gamma)O-19 reaction cross section, which is important for studies on the nucleosyntheses in the non-standard inhomogeneous big-bang model and the s process in cool giant stars, was measured by prompt gamma-ray detection methods at thermal and keV energies. In the measurement at thermal energy, ten primary and secondary gamma-ray transitions were identified and the total capture cross section was obtained as 156 +/- 16 mub, assuming a new O-19 neutron separation energy of 3.964 MeV. In the measurement at keV energies, three primary transitions were observed and the total capture cross sections were obtained as 8.5 +/- 1.7 and 11.9 +/- 1.8 mub at 42 and 65 keV, respectively.