Model-based controllers are often essential for effective control of nonlinear processes. Performance and robustness of these controllers are affected by the inevitable modeling errors, and parameter adaptation is a technique to robustify the model-based controllers. In this paper, an adaptive internal model control (AdIMC) for a class of minimum-phase input-output linearizable nonlinear systems with parameter uncertainty is presented. Internal model control (IMC) for nonlinear systems is developed directly from input-output linearization. The parameter adaptation for the IMC is based on process and model outputs, and the state variables predicted by the model only. Asymptotic tracking and convergence of unknown parameters by the proposed adaptation, is first shown theoretically. Then, AdIMC is applied to two nonlinear processes (a fermenter and a neutralization process), and its performance for a variety of disturbances and modeling errors is studied. The theoretical and simulation results show that the proposed AdIMC improves the performance and robustness of the IMC controller for nonlinear processes. Also, the proposed adaptation can easily be implemented in the IMC structure.