  The $n$-dimensional affine group over the integers is the group $\mathcal
G_n$ of all affinities on $\mathbb R^{n}$ which leave the lattice $ \mathbb
Z^{n}$ invariant. $\mathcal G_n$ yields a geometry in the classical sense of
the Erlangen Program. In this paper we construct a $\mathcal G_n$-invariant
measure on rational polyhedra in $\mathbb R^n$, i.e., finite unions of
simplexes with rational vertices in $\mathbb R^n$, and prove its uniqueness.
Our main tool is given by the Morelli-W{\l}odarczyk factorization of birational
toric maps in blow-ups and blow-downs (solution of the weak Oda conjecture).
