We consider a rate of convergence of the Levenberg-Marquardt method (LMM) for solving a system of nonlinear equations $F(x)=0$, where $F$ is a mappingfrom $R^n$ into $R^m$. It is well-known that LMM has a quadratic rate of convergence when $m=n$, the Jacobian matrix of $F$ is nonsingular at a solution $x$ and an initial point is chosen sufficiently close to $x$. In this paper, we show that if $\|F(x)\|$ provides a local error bound for the system of nonlinear equations, then a sequence generated by LMM converges to the solution set quadratically.