  A new class of $A^{(1)}_n$ integrable lattice models is presented. These are
interaction-round-a-face models based on fundamental nimrep graphs associated
with the $A^{(1)}_n$ conjugate modular invariants, there being a model for each
value of the rank and level. The Boltzmann weights are parameterized by
elliptic theta functions and satisfy the Yang-Baxter equation for any fixed
value of the elliptic nome q. At q=0, the models provide representations of the
Hecke algebra and are expected to lead in the continuum limit to coset
conformal field theories related to the $A^{(1)}_n$ conjugate modular
invariants.
