  The scaling limit of the Heisenberg XXZ spin chain at zero magnetic field is
studied in the gapped antiferromagnetic phase. For a spin-chain ring having
$N_x$ sites, the universal Casimir scaling function, which characterises the
leading finite-size correction term in the large-$N_x$ expansion of the ground
state energy, is calculated by numerical solution of the nonlinear integral
equation of the convolution type. It is shown, that the same scaling function
describes the temperature dependence of the free energy of the infinite XXZ
chain at low enough temperatures in the gapped scaling regime.
