  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 $\leq \nu_0\leq
10^9$ 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}\lesssim P_C(t_0)\lesssim 7.65\times 10^{-7}$ or Stokes
circular polarization parameter $2.7 \times 10^{-13}$ K $\lesssim
|V(t_0)|\lesssim 2 \times 10^{-6}$ K for values of the cosmic magnetic field
amplitude at present in the range $10^{-9}$ G $\lesssim B\lesssim 8\times
10^{-8}$ G. On the other hand, for not perpendicular propagation with respect
to the cosmic magnetic field we find $10^{-15}\lesssim P_C(t_0)\lesssim 6\times
10^{-12}$ or $2.72 \times 10^{-15}$ K $\lesssim |V(t_0)| \lesssim 10^{-11}$ 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
$\delta\psi_0$ due to the CM effect and constraints on the cosmic magnetic
field amplitude from current constraints on $\delta\psi_0$ due to a combination
of the CM and Faraday effects are found.
