  Tan's contact is a quantity that unifies many different properties of a
low-temperature gas with short-range interactions, from its momentum
distribution to its spatial two-body correlation function. Here, we use a
Ramsey interferometric method to realize experimentally the thermodynamic
definition of the two-body contact, i.e. the change of the internal energy in a
small modification of the scattering length. Our measurements are performed on
a uniform two-dimensional Bose gas of $^{87}$Rb atoms across the
Berezinskii-Kosterlitz-Thouless superfluid transition. They connect well to the
theoretical predictions in the limiting cases of a strongly degenerate fluid
and of a normal gas. They also provide the variation of this key quantity in
the critical region, where further theoretical efforts are needed to account
for our findings.
