We present a novel approach to the long-standing problem of locating the position of the compact nonthermal radio source Sgr A* on infrared images of the Galactic center region. Using the VLA, we have detected SiO and H2 maser emission toward several sources within the central parsec of our Galaxy. These masers arise from the innermost parts of circumstellar envelopes of giant and supergiant stars that are members of the nuclear star cluster and appear as compact infrared sources in a diffraction-limited 2.2 micron infrared image. One of the SiO masers is associated with the M-type supergiant IRS 7, the most prominent 2.2 micron point source in the Galactic center region. The radio data allow measurements of the maser positions relative to the compact non-thermal radio continuum source Sgr A* with milliarcsecond accuracy. Because stellar SiO masers near the Galactic center trace their host stars to within a few milliarcseconds, these relative positions can be used to calibrate the plate scale and rotation of the infrared image. Our method allows registration of the radio relative to the infrared reference frame with an estimated accuracy of 0.''03. Using the improved position accuracy we put a stringent upper limit on Sgr A*'s 2.2 micron flux density that is significantly lower than values predicted by recent theoretical model calculations.