  We consider models of anomaly-mediated supersymmetry breaking (AMSB) in which
the grand unification (GUT) scale is determined by the vacuum expectation value
of a chiral superfield. If the anomaly-mediated contributions to the potential
are balanced by gravitational-strength interactions, we find a
model-independent prediction for the GUT scale of order $M_{\rm Planck} /
(16\pi^2)$. The GUT threshold also affects superpartner masses, and can easily
give rise to realistic predictions if the GUT gauge group is asymptotically
free. We give an explicit example of a model with these features, in which the
doublet-triplet splitting problem is solved. The resulting superpartner
spectrum is very different from that of previously considered AMSB models, with
gaugino masses typically unifying at the GUT scale.
