A new method is proposed for semiglobally stabilizing a large class of nonlinear systems. Assuming the nonlinearities satisfy a Lipschitz condition on a prescribed compact set is sufficient to ensure that they can be characterized using conic sector bounds. These conic sector bounds provide an uncertain linear system description of the original nonlinear system. A stabilizing linear feedback controller for this uncertain linear system is sought using an established method from robust control. Finally, an iterative scheme is proposed to ensure that the trajectories of the stabilized system remain inside the prescribed compact set. The major advantage of this method is that we only need crude information about the system nonlinearities, since we do not require the controller to cancel them. The resulting Linear feedback controller is easily implemented, and yields smooth stabilized trajectories, which are favored on physical grounds.