This paper presents a derivation of the optimum convergence constant to yield the most rapid convergence under a desired mean-square error (MSE) for echo canceller using the LMS algorithm. For white input data, the optimum convergence constant is a simple closed-form function of the number of filter taps, the input signal variance, the initial MSE, and the desired MSE. This characteristic makes it easily designed in many practical applications. Computer simulations are also employed to show the correctness and effectiveness of the derived results.