We present observations of ro-vibrational OH and CO emission from the Herbig Be star HD 100546. The emission from both molecules arises from the inner region of the disk extending from approximately 13 AU from the central star. The velocity profiles of the OH lines are narrower than the velocity profile of the [O I] 6300 Angstrom line indicating that the OH in the disk is not cospatial with the O I. This suggests that the inner optically thin region of the disk is largely devoid of molecular gas. Unlike the ro-vibrational CO emission lines, the OH lines are highly asymmetric. We show that the average CO and average OH line profiles can be fit with a model of a disk comprised of an eccentric inner wall and a circular outer disk. In this model, the vast majority of the OH flux (75%) originates from the inner wall, while the vast majority of the CO flux (65%) originates on the surface of the disk at radii greater than 13 AU. Eccentric inner disks are predicted by hydrodynamic simulations of circumstellar disks containing an embedded giant planet. We discuss the implications of such a disk geometry in light of models of planet disk tidal interactions and propose alternate explanations for the origin of the asymmetry.