We present a simple and computationally efficient algorithm, based on the accelerated Newton's method, to solve the root finding problem associated with the projection onto the ℓ1-ball problem. Considering an interpretation of the Michelot's algorithm as Newton method, our algorithm can be understood as an accelerated version of the Michelot's algorithm, that needs significantly less major iterations to converge to the solution. Although the worst-case performance of the propose algorithm is O(n2), it exhibits in practice an O(n) performance and it is empirically demonstrated that it is competitive or faster than existing methods.