This paper presents an efficient algorithm for computing the capacity of discrete memoryless channels. The algorithm uses Newton's method which is known to be quadratically convergent. First, a system of nonlinear equations termed Kuhn-Tucker equations is formulated, which has the capacity as a solution. Then Newton's method is applied to the Kuhn-Tucker equations. Since Newton's method does not guarantee global convergence, a continuation method is also introduced. It is shown that the continuation method works well and the convergence of the Newton algorithm is guaranteed. By numerical examples, effectiveness of the algorithm is verified. Since the proposed algorithm has local quadratic convergence, it is advantageous when we want to obtain a numerical solution with high accuracy.