This paper deals with adaptive control of a class of nonlinear dynamic systems with nonlinear parametrization. In this class, the state variables are assumed to be accessible and the nonlinearity in the parameters is assumed to be either convex or concave. By introducing a tuning function and an adaptive law based on a min-max strategy, it is shown that such a class of dynamic systems can be adaptively controlled in a stable manner. Global boundedness of the overall adaptive system and tracking to within a desired precision are established with the new adaptive controller. The proposed controller is applied to a precise-positioning problem in the presence of nonlinearly parametrized friction dynamics. It is shown that the controller leads to position errors and friction compensation that are orders of magnitude better than those based on estimation of linear parametrizations. The fact that the new parameter estimation strategy used is distinct from the traditionally used gradient schemes, permits the expansion of the scope of adaptive control, which has been restricted hitherto, in most cases, to systems with linear parametrization.