In this paper, based on an alternating direction method of multipliers (ADMM), a novel distributed algorithm is proposed to address the economic dispatch problem (EDP) in islanded microgrids. Unlike most of the existing studies that investigate the EDP with quadratic cost functions, the algorithm proposed in this paper is able to solve the EDP with general convex cost functions. Moreover, by using the center-free algorithm and the projection method, the traditional centralized ADMM is extended to the distributed implementation framework, and a fully distributed solution to the EDP is obtained. Furthermore, the proposed algorithm can deal with both equality and inequality constraints in the EDP, and the supply-demand balance can be guaranteed at any time provided that the sum of initial power outputs is equal to the total demand. The convergence property of the proposed algorithm is strictly proved. Finally, some examples are provided to further demonstrate the effectiveness of the proposed algorithm.