A muon beam polarimeter was built for the SMC experiment at the CERN SPS, for beam energies of 100 and 190 GeV. The beam polarization is determined from the asymmetry in the elastic scattering off the polarized electrons of a ferromagnetic target whose magnetization is periodically reversed. At muon energies of 100 and 190 GeV the measured polarization is Pμ = -0.80±0.03 (stat.)±0.02 (syst.) and Pμ = -0.797±0.011 (stat.)±0.012 (syst.), respectively. These results agree with measurements of the beam polarization using a shape analysis of the decay positron energy spectrum.