  Mn$_2$Au is an important antiferromagnetic (AF) material for spintronics
applications. Due to its very high N\'eel temperature of about 1500 K, some of
the basic properties are difficult to explore, such as the AF susceptibility
and the exchange constants. Experimental determination of these properties is
further complicated in thin films by unavoidable presence of uncompensated and
quasiloose spins on antisites and at interfaces. Using x-ray magnetic circular
dichroism (XMCD), we have measured the spin and orbital contribution to the
susceptibility in the direction perpendicular to the in-plane magnetic moments
of a Mn$_2$Au(001) film and in fields up to 8 T. By performing these
measurements at a low temperature of 7 K and at room temperature, we were able
to separate the loose spin contribution from the susceptibility of AF coupled
spins. The value of the AF exchange constant obtained with this method for a 10
nm thick Mn$_2$Au(001) film equals to (24 $\pm$ 5) meV.
