In this paper, we focus on an asynchronous distributed optimization problem. In our problem, each node is endowed with a convex local cost function, and is able to communicate with its neighbors over a directed communication network. Furthermore, we assume that the communication channels between nodes have limited bandwidth, and each node suffers from processing delays. We present a distributed algorithm which combines the Alternating Direction Method of Multipliers (ADMM) strategy with a finite time quantized averaging algorithm. In our proposed algorithm, nodes exchange quantized valued messages and operate in an asynchronous fashion. More specifically, during every iteration of our algorithm each node (i) solves a local convex optimization problem (for the one of its primal variables), and (ii) utilizes a finite-time quantized averaging algorithm to obtain the value of the second primal variable (since the cost function for the second primal variable is not decomposable). We show that our algorithm converges to the optimal solution at a rate of O (1/ k) (where k is the number of time steps) for the case where the local cost function of every node is convex and not-necessarily differentiable. Finally, we demonstrate the operational advantages of our algorithm against other algorithms from the literature.