Generation of cosmic microwave background (CMB) elliptic polarization due to the Cotton–Mouton (CM) effect in a cosmic magnetic field is studied. We concentrate on the generation of CMB circular polarization and on the rotation angle of the CMB polarization plane from the decoupling time until at present. For the first time, a rather detailed analysis of the CM effect for an arbitrary direction of the cosmic magnetic field with respect to photon direction of propagation is done. Considering the CMB linearly polarized at the decoupling time, it is shown that the CM effect is one of the most substantial effects in generating circular polarization especially in the low part of the CMB spectrum. It is shown that in the frequency range 10 8 Hz ≤ ν≤ 10 9 Hz, the degree of circular polarization of the CMB at present for perpendicular propagation with respect to the cosmic magnetic field is in the range 10 - 13 ≲ P C (t) ≲ 7.65 × 10 - 7 or Stokes circular polarization parameter 2.7 × 10 - 13 K ≲ | V(t) | ≲ 2 × 10 - 6 K for values of the cosmic magnetic field amplitude at present in the range 10 - 9 G ≲ B≲ 8 × 10 - 8 G. On the other hand, for not perpendicular propagation with respect to the cosmic magnetic field we find 10 - 15 ≲ P C (t) ≲ 6 × 10 - 12 or 2.72 × 10 - 15 K ≲ | V(t) | ≲ 10 - 11 K, for the same values of the cosmic magnetic field amplitude and same frequency range. Estimates on the rotation angle of the CMB polarization plane δψ due to the CM effect and constraints on the cosmic magnetic field amplitude from current constraints on δψ due to a combination of the CM and Faraday effects are found.