This paper considers the design of a robust adaptive H-infinity optimal controller based on the IMC structure. Specific attention is Focused on the practically important class of stable plants with one open right half-plane zero, The certainty equivalence principle of adaptive control is used to combine a robust adaptive law with a robust H-infinity internal model controller to obtain an adaptive H-infinity internal model control scheme with provable guarantees of stability and robustness. The approach used earlier for designing and analyzing adaptive H-2 optimal control schemes is seen to carry over in a natural fashion to the adaptive H-infinity optimal control case. This attests to the generality of that approach, which not only provides a theoretical basis for analytically justifying some of the reported industrial successes of existing adaptive internal model control schemes, but also opens up the possibility of synthesizing new ones by simply combining a robust adaptive law with a robust internal model controller structure.