We study the effect of Faraday rotation due to a uniform magnetic field on the polarization of the cosmic microwave background. Scalar fluctuations give rise only to parity-even E-type polarization of the cosmic microwave background. However in the presence of a magnetic field, a nonvanishing parity-odd B-type polarization component is produced through Faraday rotation. We derive the exact solution for the E and B modes generated by scalar perturbations including the Faraday rotation effect of a uniform magnetic field, and evaluate their cross correlations with temperature anisotropies. We compute the angular autocorrelation function of the B-modes in the limit that the Faraday rotation is small. We find that uniform primordial magnetic fields of present strength around B0=10-9 G rotate E-modes into B-modes with amplitude comparable to those due to the weak gravitational lensing effect at frequencies around ν=30 GHz. The strength of B-modes produced by Faraday rotation scales as B0/ν2. We evaluate also the depolarizing effect of Faraday rotation upon the cross correlation between temperature anisotropy and E-type polarization.