An important unsolved problem in nonlinear control is that of stabilization with a prescribed region of stability. In this paper, sufficient conditions are obtained for the existence of a linear feedback stabilizing an equilibrium point of a given nonlinear system with the resulting region of asymptotic stability (RAS) containing a ball of given radius. Conditions for global stabilization are also given. Feedback stabilization is achieved while satisfying a certain robustness property. The technique is applied to planar systems, resulting in a complete design methodology for this case. Examples and simulations illustrating the method are presented.