  We extend the results of Bey, Hen, and Wills
(http://arxiv.org/abs/math/0606089). In this paper, we show that, up to
equivalence under unimodular transformations, there is exactly one class of
$d$-simplices having $k \ge 1$ interior lattice points and minimal volume
$\frac{1}{d!}(dk+1)$.
