  The arithmetic motivic Poincar\'e series of a variety $V$ defined over a
field of characteristic zero, is an invariant of singularities which was
introduced by Denef and Loeser by analogy with the Serre-Oesterl\'e series in
arithmetic geometry. They proved that this motivic series has a rational form
which specializes to the Serre-Oesterl\'e series when $V$ is defined over the
integers. This invariant, which is known explicitly for a few classes of
singularities, remains quite mysterious. In this paper we study this motivic
series when $V$ is an affine toric variety. We obtain a formula for the
rational form of this series in terms of the Newton polyhedra of the ideals of
sums of combinations associated to the minimal system of generators of the
semigroup of the toric variety. In particular, we deduce explicitly a finite
set of candidate poles for this invariant.
