To estimate unmeasured states in dynamical systems with parametric uncertainties one can use adaptive observers. In this technical note a new method for adaptive redesign of reduced order observers for nonlinear systems is presented. The redesign makes the observer robust as it guarantees that the state estimation error converges to zero in the presence of parametric uncertainties. To enable parameter adaption, a new type of update law is derived using Lyapunov analysis and a nonlinear coordinate transformation. The uniqueness of the approach is that it allows terms containing both uncertainty and nonlinearity in the unmeasured states to appear in the dynamics of the unmeasured states, and still achieves convergence of the state estimate without requiring persistent excitation. Two examples are presented that demonstrate these properties.