As part of The HST/ACS Survey of Galactic Globular Cluster, we have imaged the globular clusters that are canonical members of the Sagittarius dSph galaxy (Sgr), including M54, which is either in or along the line of sight of the core of the Sagittarius dSph. The M54 field CMD, which extends to more than five magnitudes below the M54 main sequence turn-off (MSTO), clearly shows multiple RGBs and a complex array of other features, especially at the Sgr/M54 MSTO. We revise the parameters of the other nominal Sgr member clusters and the Sgr core using ACS main sequence fiducials as well as the new stellar evolution isochrones constructed specifically for the ACS survey. The distance to the Sgr core is a key constraint on theoretical models of the tidal disruption of the dSph and the distance scale of its extensive star streams. Our MS-fitting to the four central Sgr globular clusters provides one set of distance estimates to the Sgr system. We also identify five Milky Way globular clusters in/near the Milky Way bulge -NGC 6624, NGC 6809, NGC 6681, NGC 6637 and NGC 6652 -for which our CMDs also include the main sequence of the central body of Sagittarius at fainter magnitudes. Thus, we can also measure the relative distance moduli between Sgr and these generally well-studied foreground bulge clusters. Through both the direct MS-fitting and relative MS comparisons to foreground clusters, we find that the Sgr core is likely to be some 20-30% farther than the canonical distances derived or used in many earlier investigations of the Sgr system. Support for this work (GO-10775) was provided by NASA through a grant from the Space Telescope Science Institute which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.