We show that the ground states of the three-dimensional XXZ Heisenberg ferr
omagnet with a III interface have excitations localized in a subvolume of l
inear size R with energies bounded by O(1/R-2). As part of the proof we sho
w the equivalence of ensembles for the III interface states in the followin
g sense: In the thermodynamic limit the states with fixed magnetization yie
ld the same expectation values for gauge invariant local observables as a s
uitable grand canonical state with fluctuating magnetization. Here, gauge i
nvariant means commuting with the total third component of the spin, which
is a conserved quantity of the Hamiltonian. As a corollary of equivalence o
f ensembles we also prove the convergence of the thermodynamic limit of seq
uences of canonical states (i.e., with fixed magnetization).