We report here the detection of circular polarization in the Galactic Center black hole candidate , Sagittarius A* . The detection was made at 4.8 GHz and 8.4 GHz with the Very Large Array . We find that the fractional circular polarization at 4.8 GHz is m _ { c } = -0.36 \pm 0.05 \% and that the spectral index of the circular polarization is \alpha = -0.6 \pm 0.3 ( m _ { c } \propto \nu ^ { \alpha } ) . The systematic error in m _ { c } is less than 0.04 % at both frequencies . In light of our recent lower limits on the linear polarization in Sgr A* , this detection is difficult to interpret with standard models . We consider briefly whether scattering mechanisms could produce the observed polarization . Detailed modeling of the source and the scattering medium is necessary . We propose a simple model in which low energy electrons reduce linear polarization through Faraday depolarization and convert linear polarization into circular polarization . Circular polarization may represent a significant new parameter for studying the obscured centimeter wavelength radio source in Sgr A* .