We report low-energy (0.001 -10-eV) electron-CO scattering cross sections obtained using an exactexchange (via a separable-exchange formulation) plus a parameter-free correlation-polarization model in the fixed-nuclei approximation (FNA). The differential, total, and momentum-transfer cross sections are reported for rotationally elastic, inelastic, and summed processes. To remove the limitations of the FNA with respect to the convergence of total and differential cross sections, the multipole-extractedadiabatic-nuclei approximation is used. The position and width of the well-known H shape-resonance structure in the cross section around 2 eV are reproduced quite well; however, some discrepancy between theory and experiment in the magnitude of the total cross section in the resonance region exists. We also present results for H shape-resonance parameters as a function of internuclear separation. Differential-cross-section results agree well with the measurements of Tanaka, Srivastava, and Chutjian [J.Chem. Phys. 69, 5329 (1978)]but are about a factor of 2 larger than the results obtained by Jung et al. [J.Phys. B 15, 3535 (1982)] in the vicinity of the 'II resonance.PACS number(s): 34.80.Bm, 34.80.Gs