A general expression for the probability of photon emission by an electron in a unidirectional bichromatic classical field is calculated making use of the exact solution to the Dirac equation for the electron in the field. In the limit that one component of the field is weak, a cross section for scattering a photon out of this component is derived. This has a complicated form but numerical values have been obtained in the particular case of an electron of energy 5 GeV travelling through radiation from a ruby laser (considered weak) and an intense CO$sub 2$ laser. It is found that, as a function of the intensity of the CO$sub 2$ laser, the cross section first increases, but eventually decreases to values below the Compton cross section. The spectrum of the scattered radiation is also calculated; is shows intensity-dependent shifts attributable to a field-induced drift velocity of the electron.