  We have reconstructed possible orbits for a collection of stars located
within 0.5 arcsec of Sgr A*. These orbits are constrained by observed stellar
positions and angular proper motions. The construction of such orbits serves as
a baseline from which to search for possible deviations due to the unseen mass
distribution in the central 1000 AU of the Galaxy. We also discuss the
likelihood that some of these stars may eventually exhibit detectable
relativistic effects, allowing for interesting tests of general relativity
around the 2.6 x 10^6 solar mass central object.
