  We solve in closed form the simplest (su(1|1)) supersymmetric version of
Inozemtsev's elliptic spin chain, as well as its infinite (hyperbolic)
counterpart. The solution relies on the equivalence of these models to a system
of free spinless fermions, and on the exact computation of the Fourier
transform of the resulting elliptic hopping amplitude. We also compute the
thermodynamic functions of the finite (elliptic) chain and their low
temperature limit, and show that the energy levels become normally distributed
in the thermodynamic limit. Our results indicate that at low temperatures the
su(1|1) elliptic chain behaves as a critical XX model, and deviates in an
essential way from the Haldane-Shastry chain.
