  Given positive integers e and s we consider Gorenstein Artinian local rings R
of embedding dimension e whose maximal ideal $\mathfrak{m}$ satisfies
$\mathfrak{m}^s\ne 0=\mathfrak{m}^{s+1}$. We say that R is a compressed
Gorenstein local ring when it has maximal length among such rings. It is known
that generic Gorenstein Artinian algebras are compressed. If $s\ne 3$, we prove
that the Poincare series of all finitely generated modules over a compressed
Gorenstein local ring are rational, sharing a common denominator. A formula for
the denominator is given. When s is even this formula depends only on the
integers e and s. Note that for $s=3$ examples of compressed Gorenstein local
rings with transcendental Poincare series exist, due to B{\o}gvad.
