This paper considers the average-consensus problem in a network with arbitrary (but finite) communication delays. A novel Distributed-Averaging (DA) algorithm is presented and shown to achieve average-consensus if at any time t there exists a finite time interval [t, T_t] over which each node can communicate (via a time-respecting path) with all other nodes. For consensus variables with dimensions on the order of the network size, the DA algorithm requires an order of magnitude less data to be communicated and stored at each node as compared to an idealized algorithm that floods the initial data. In the companion paper, practical applications of the DA algorithm are provided along with numerical examples.