The time-step in integration process has two restri ctions. The first one is the time step restriction due to accuracy requirement τac and the second one is the time-step restriction du e to stability requirement τst . The most of explicit methods have small stability regio ns and consequently small τst . It obliges us to solve stiff problems with small step size τst << τac . The implicit methods work well with stiff problem s but these methods require more work per step than the explici t methods. In this study, a non-linear absolutly st able explicit one step numerical integration algorithm i s proposed for solving non linear stiff initial-val ue problems in ordinary differential equations. The al gorithm is based on deriving a non-linear relation between the dependent variable and its derivatives from the well known Taylor expansion. The accuracy of the method depends on some unknown parameter inserted in Taylor expansion and determined from the error analysis. The accuracy and stability properties of the method are investigated and shown to yield at least third order and A-stable. The results obtained in the numerical exp eriments show the efficiency of the present method in solving stiff initial value problems.