Recent measurements, of the velocities of stars near the center of the Milky Way have provided the strongest evidence for the presence of a supermassive black hole in a galaxy, but the observational uncertainties poorly constrain many of the properties of the black hole. Determining the accelerations of stars in their orbits around the center provides much more precise information about the position and mass of the black hole. Here we report measurements of the accelerations for three stars located 0.005 pc from the central radio source Sgr A*; these accelerations are comparable to those experienced by the Earth as it orbits the Sun. These data increase the inferred minimum mass density in the central region of the Galaxy by an order of magnitude relative to previous results and localized the dark mass to within 0.05 +- 0.04 arcsec of the nominal position of Sgr A*. In addition, the orbital period of one of the observed stars could be as short as 15 years, allowing us the opportunity in the near future to observe an entire period.