The Sagittarius dwarf spheroidal galaxy, the closest satellite galaxy of the Milky Way, has survived for many orbits about the Galaxy. Extent numerical calculations modeled this galaxy as a system with a centrally-concentrated mass profile, following the light, and found that it should lose more than one-half of its mass every 2-4 orbits and be completely disrupted long before now. Apparently the Sagittarius dwarf spheroidal, and by implication other dSph galaxies, do not have a centrally-concentrated profile for their dark matter. We develop a model in which the stars of the Sgr dwarf are embedded in a constant-density dark matter halo, representing the core of a tidally-limited system, and show that this is consistent with its survival. We present new photometric and kinematic observations of the Sagittarius dwarf spheroidal and show these data are consistent with this explanation for the continued existence of this galaxy. The Sagittarius dwarf is being tidally distorted and is tidally limited, but is not disrupted as yet. The corresponding minimum total mass is 10^9^M_{sun}_, while the central mass to visual light ratio is ~50 in Solar units. Our new photographic photometry allows the detection of main-sequence stars of the Sagittarius dwarf over an area of 22x8{deg}. The Sagittarius dwarf is prolate, with axis ratios ~3:1:1. For an adopted distance of 16+/-2kpc from the Galactic center on the opposite side of the Galaxy to the Sun, the major axis is >~9kpc long and is aligned approximately normal to the plane of the Milky Way Galaxy, roughly following the coordinate line l=5{deg}. The central velocity dispersion of giant stars which are members of the Sagittarius dwarf is 11.4+/-0.7km/s and is consistent with being constant over the face of the galaxy. The gradient in mean line-of-sight velocity with position along the major axis, dv/db, is ~0km/s/degree in the central regions and increases in amplitude to dv/db=-3km/s/degree over the outermost three degrees for which we have data. A first measurement of the proper motion of the Sagittarius dwarf determines the component of its space velocity parallel to its major axis to be 250+/-90km/s, directed towards the Galactic Plane. We model these kinematic data to determine the orbit of the Sagittarius dwarf. Our best fit model has an orbital period of <~1Gyr and has the Sagittarius dwarf spheroidal close to perigalacticon. This period is shorter, by about a factor of >~10, than the age of the bulk of its stellar population. (Copyright) 1997 American Astronomical Society.