In this paper, the problem of stabilization of unknown nonlinear dynamical systems is considered. An adaptive feedback law is constructed that is based on the switching adaptive strategy proposed by the author and uses linear-in-the-weights neural networks accompanied with appropriate robust adaptive laws in order to estimate the time-derivative of the control Lyapunov function (CLF) of the system. The closed-loop system is shown to be stable; moreover, the state vector of the controlled system converges to a ball centered at the origin and having a radius that can be made arbitrarily small by increasing the high gain K and the number of neural network regressor terms. No growth conditions on the nonlinearities of the system are imposed with the exception that such nonlinearities are sufficiently smooth. Finally, we mention that neither the system dynamics or the CLF of the system need to be known in order to apply the proposed methodology.