In 1918, Harlow Shapley first noted that globular clusters were concentrated toward the constellation of Sagittarius, and hence the Sun was not near the center of the Milky Way. Since that time astronomers have expended considerable effort to determine the distance to the center of the Milky Way, because any change in the value of this distance, R0, has a widespread impact on astronomy and astrophysics. Beginning in 1991, we have conducted observations with the VLBA designed to make possible a program to measure the distance to the Galactic Center via a trigonometric parallax. This could be accomplished with the VLBA using Sgr A∗ as a phase reference for one or more (weaker) compact extragalactic sources. A time series of measurements of the position of Sgr A∗ relative to an extragalactic source should show the effects of the annual ≈ ±0.12 mas signature of the Earth's orbit around the Sun (trigonometric parallax), as well as the ≈ 6 mas yr−1 secular motion caused by the Sun's orbit around the Galactic Center.