  Power-counting arguments based on extended superfields have been used to
argue that two-dimensional supersymmetric sigma models with (4,0) supersymmetry
are finite. This result is confirmed up to three loop order in pertubation
theory by an explicit calculation using (1,0) superfields. In particular, it is
shown that the finite counterterms which must be introduced into the theory in
order to maintain (4,0) supersymmetry are precisely the terms that are required
to establish ultra-violet finiteness.
