  Let $T$ be the subgroup of diagonal matrices in the group SL(n). The aim of
this paper is to find all finite-dimensional simple rational SL(n)-modules $V$
with the following property: for each point $v\in V$ the closure $\bar{Tv}$ of
its $T$-orbit is a normal affine variety. Moreover, for any SL(n)-module
without this property a $T$-orbit with non-normal closure is constructed. The
proof is purely combinatorial: it deals with the set of weights of simple
SL(n)-modules. The saturation property is checked for each subset in the set of
weights.
