The radio source Sagittarius A* at the center of our Galaxy is believed to be a 2.6 x 10(6) M. black hole that accretes gas from the winds of nearby stars. We show that limits on the X-ray and infrared emission from the Galactic center provide an upper limit of similar to 8 x 10(-5) M. yr(-1) on the mass accretion rate in Sgr A*. The advection-dominated accretion flow (ADAF) model favors a rate of less than or similar to 10(-5) M. yr(-1). In comparison, the Bondi accretion rate onto Sgr A*, estimated using the observed spatial distribution of mass-losing stars and assuming noninteracting stellar winds, is similar to 3 x 10(-5) M. yr(-1). Thus, there is rough agreement between the Bondi, the ADAF, and the X-ray-inferred accretion rates for Sgr A*. We discuss uncertainties in these estimates, emphasizing the importance of upcoming observations by the Chandra X-ray Observatory for tightening the X-ray-derived limits.