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This paper is concerned with the distributed averaging problem subject to a quantization constraint. Given a group of agents associated with scalar numbers, it is assumed that each pair of agents can communicate with each other with a prescribed probability, and that the data being exchanged between them is quantized. In this part of the paper, it is proved that the stochastic gossip algorithm proposed in a recent paper leads to reaching the quantized consensus. Some important properties of the system in the steady-state (after reaching the consensus) are also derived. The results developed here hold true for any arbitrary quantization, provided the tuning parameter of the gossip algorithm is chosen properly. The expected value of the convergence time bounded in the second part of the paper.