In this paper, we propose a fully distributed approximate message passing (AMP) algorithm, which reconstructs an unknown vector from its linear measurements obtained at nodes in a network. The proposed algorithm is a distributed implementation of the centralized AMP algorithm, and consists of the local computation at each node and the global computation using communications between nodes. For the global computation, we propose a distributed algorithm named summation propagation to calculate a summation required in the AMP algorithm. The proposed distributed AMP algorithm does not require any central node such as a fusion center, and can be realized only with locally available information at each node. Simulation results show that the proposed algorithm can achieve the same estimation accuracy as that of the centralized AMP algorithm.