In this paper a new general method is developed by means of which one can ascertain whether a nonlinear autonomous system is asymptotically stable. The method is essentially an extension to nonlinear systems of a theorem developed earlier by the first author for linear autonomous systems. Necessary and sufficient conditions are specified, the satisfaction of which guarantees that the system being studied is asymptotically stable. The new method, by design, always uses a positive-definite function which satisfies LyapunovÂ¿s stability theorem. However, the new method uses only one positive-definite function, in contrast to LyapunovÂ¿s stability theorem, which requires two functions to be definite at the same time. In addition, the new method specifies the stability function at the outset once a mathematical system model has been obtained.