The author studies the inelastic scattering, accompanied by the transfer of L photons, of fast electrons by the CO molecule (in the 1 Sigma + state) in the presence of an infra-red laser beam taken in the dipole approximation. Both the laser-projectile and the laser-molecule interactions are treated non-perturbatively while the electron-molecule interaction is treated within the first Born approximation. The utility of the author's formalism is illustrated by computing the cross sections for vibration-rotation transitions, vj to v'j', in the CO molecule due to electron impact for the absorption of L photons. The author discusses also the influence of various laser parameters (polarization, frequency and intensity) which strongly affect the dynamics of the collision. The results are also compared with those for the field-free case.