  We investigate whether the (planar, two complex scalar) dilatation operator
of N=4 gauge theory can be, perturbatively and, perhaps, non-perturbatively,
described by an integrable long range spin chain with elliptic exchange
interaction. Such a chain was introduced some time ago by Inozemtsev. In the
limit of sufficiently ``long'' operators a Bethe ansatz exists, which we apply
at the perturbative two- and three-loop level. Spectacular agreement is found
with spinning string predictions of Frolov and Tseytlin for the two-loop
energies of certain large charge operators. However, we then go on to show that
the agreement between perturbative gauge theory and semi-classical string
theory begins to break down, in a subtle fashion, at the three-loop level. This
corroborates a recently found disagreement between three-loop gauge theory and
near plane-wave string theory results, and quantitatively explains a previously
obtained puzzling deviation between the string proposal and a numerical
extrapolation of finite size three-loop anomalous dimensions. At four loops and
beyond, we find that the Inozemtsev chain exhibits a generic breakdown of
perturbative BMN scaling. However, our proposal is not necessarily limited to
perturbation theory, and one would hope that the string theory results can be
recovered from the Inozemtsev chain at strong 't Hooft coupling.
