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 mc=-0.36 +/- 0.05% and that the spectral index of the circular polarization is alpha =-0.6 +/- 0.3 (mc \propto nu alpha ). The systematic error in mc 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*.