  We present a high-precision determination of the universal contact parameter
in a strongly interacting Fermi gas. In a trapped gas at unitarity we find the
contact to be $3.06 \pm 0.08$ at a temperature of 0.08 of the Fermi temperature
in a harmonic trap. The contact governs the high-momentum (short-range)
properties of these systems and this low temperature measurement provides a new
benchmark for the zero temperature homogeneous contact. The experimental
measurement utilises Bragg spectroscopy to obtain the dynamic and static
structure factors of ultracold Fermi gases at high momentum in the unitarity
and molecular Bose-Einstein condensate (BEC) regimes. We have also performed
quantum Monte Carlo calculations of the static properties, extending from the
weakly coupled Bardeen-Cooper-Schrieffer (BCS) regime to the strongly coupled
BEC case, which show agreement with experiment at the level of a few percent.
