Recently, Yamashita and Fukushima [11] established an interesting quadratic convergence result for the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima by using _k=||F(x_k)||, where [1,2], instead of _k=||F(x_k)||² as the Levenberg-Marquardt parameter. If ||F(x)|| provides a local error bound for the system of nonlinear equations F(x)=0, it is shown that the sequence {x_k} generated by the new method converges to a solution quadratically, which is stronger than dist(x_k,X^*)0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.