A nonrelativistic treatment is given of electron-electron scattering in the presence of a laser field. The field is accounted for by the external field approximation and is represented by a circularly polarised monochromatic plane-wave field. A simple analytic expression is derived for the transition amplitude, which is shown to exhibit internal resonances as well as intensity-dependent shifts. The former is the nonrelativistic limit of the resonant Moller scattering predicted previously by Oleinik (1967). The latter, however, appears in a higher order of v/c and is consequently negligible for very slow electrons. The differential cross section of the scattering is also given where the effect of the spin and symmetry is teken into account explicitly. The width of resonances is introduced phenomenologically but its connection with previous methods is established. Consideration is also given to the experimental conditions under which the effects may become observable.