  We present the deepest near-infrared (ZJKs) photometry yet obtained of the
Sagittarius dwarf spheroidal (Sgr dSph), using VISTA to survey 11 square
degrees centred on its core. We list locations and ZJKs-band magnitudes for
over 2.9 million sources in the field. We discuss the isolation of the Sgr dSph
from the foreground and Galactic Bulge populations, identify the Sgr dSph's
horizontal branch in the near-infrared for the first time, and map the density
of the galaxy's stars. We present isochrones for the Sgr dSph and Bulge
populations. These are consistent with the previously-reported properties of
the Sgr dSph core: namely that it is dominated by a population between [Fe/H] ~
-1 dex and solar, with a significant [alpha/Fe] versus [Fe/H] gradient. While
strong contamination from the Galactic Bulge prevents accurate measurement of
the (Galactic) north side of the Sgr dSph, the dwarf galaxy can be
well-approximated by a roughly ovaloid projection of characteristic size 4 x 2
degrees, beyond which the projected stellar density is less than half that of
the region surrounding the core. The galaxy's major axis is perpendicular to
the Galactic Plane, as in previous studies. We find slight evidence to confirm
a metallicity gradient in the Sgr dSph and use isochrones to fit a distance of
24.3 +/- 2.3 kpc. We were unable to fully constrain the metallicity
distribution of the Sgr dSph due to the Bulge contamination and strong
correlation of [alpha/Fe] with metallicity, however we find that metal-poor
stars ([Fe/H] <~ --1) make up <~29 per cent of the Sgr dSph's upper-RGB
population. The Bulge population is best fit by a younger population with
[Fe/H] ~ 0 and [alpha/Fe] ~ 0 or slightly higher. We find no evidence for a
split, peanut- or X-shaped Bulge population in this line of sight (l = 5.6 +/-
~1 deg, b = -14.1 +/- ~3 deg).
