The problem of estimating a sparse channel, i.e. a channel with a few non-zero taps, appears in various areas of communications. Recently, we have developed an algorithm based on iterative alternating minimization which iteratively detects the location and the value of the taps. This algorithms involves an approximate Maximum A Posteriori (MAP) probability scheme for detection of the location of taps, while a least square method is used for estimating the values at each iteration. In this work, based on the method of factor graphs and message passing algorithms, we will compute an exact solution for the MAP estimation problem. Indeed, we first find a factor graph model of this problem, and then perform the well-known min-sum algorithm on the edges of this graph. Consequently, we will find an exact estimator for the MAP problem that its complexity grows linearly with respect to the channel memory. By substituting this estimator in the mentioned alternating minimization method, we will propose an estimator that will nearly achieve the Cramer-Rao bound of the genie-aided estimation of sparse channels (estimation based on knowing the location of non-zero taps of the channel), while it can perform faster than most of the proposed algorithms in literature.