  We show that the presence of an anomalous $\rm U_A(1)$ factor in the gauge
group of string-derived models may have the new and important phenomenological
consequence of allowing the vanishing of ${\rm Str}\,{\cal M}^2$ in the
``shifted" vacuum, that results in the process of cancelling the anomalous $\rm
U_A(1)$. The feasibility of this effect seems to be enhanced by a vanishing
vacuum energy, and by a ``small" value of ${\rm Str}\,{\cal M}^2$ in the
original vacuum. In the class of free-fermionic models with vanishing vacuum
energy that we focus on, a necessary condition for this mechanism to be
effective is that ${\rm Str}\,{\cal M}^2>0$ in the original vacuum. A vanishing
${\rm Str}\,{\cal M}^2$ ameliorates the cosmological constant problem and is a
necessary element in the stability of the no-scale mechanism.
