  Let G be a complex reductive algebraic group. Fix a Borel subgroup B of G,
with unipotent radical U, and a maximal torus T in B with character group X(T).
Let S be a submonoid of X(T) generated by finitely many dominant weights. V.
Alexeev and M. Brion introduced a moduli scheme M_S which classifies pairs
(X,f) where X is an affine G-variety and f is a T-equivariant isomorphism
between the categorical quotient of X by U and the toric variety determined by
S. In this paper, we prove that M_S is isomorphic to an affine space when S is
the weight monoid of a spherical G-module with G of type A.
