  In the first two papers of this series, we determined the position of Sgr A*
on infrared images, by aligning the positions of red giant stars with positions
measured at radio wavelengths for their circumstellar SiO masers. In this
paper, we report detections of 5 new stellar SiO masers within 50" (2 pc) of
Sgr A* and new and/or improved positions and proper motions of 15 stellar SiO
masers. The current accuracies are ~1 mas in position and ~0.3 mas/y in proper
motion. We find that the proper motion of the central stellar cluster with
respect to Sgr A* is less than 45 km/s. One star, IRS 9, has a
three-dimensional speed of ~370 km/s at a projected distance of 0.33 pc from
Sgr A*. If IRS 9 is bound to the inner parsec, this requires an enclosed mass
that exceeds current estimates of the sum of the mass of Sgr A* and luminous
stars in the stellar cusp by ~0.8 x 10^6 Msun. Possible explanations include i)
that IRS 9 is not bound to the central parsec and has "fallen" from a radius
greater than 9 pc, ii) that a cluster of dark stellar remnants accounts for
some of the excess mass, and/or iii) that Ro is considerably greater than 8
kpc.
