The Routh-Hurwitz Criterion is closely related to the stability of a continuous-time system of differential equations. Its main concern is the determination of the number of zeros of the characteristic polynomial of the system with positive or negative real parts and the number of imaginary zeros. We propose a new algorithm to achieve this goal. The new technique is extremely simple to apply since it only requires elementary row operations of an associated determinant. The asymptotic stability of the system falls out as a special case of the newly proposed algorithm. Several randomly chosen examples are given to illustrate the simplicity and the effectiveness of the novel approach.