We consider an unconstrained distributed optimization problem and assume that the bit rate of the communication in the network is limited. We propose a distributed optimization algorithm with an iteratively refining quantization design, which bounds the quantization errors and ensures convergence to the global optimum. We present conditions on the bit rate and the initial quantization intervals for convergence, and show that as the bit rate increases, the corresponding minimum initial quantization intervals decrease. We prove that after imposing the quantization scheme, the algorithm still provides a linear convergence rate, and furthermore derive an upper bound on the number of iterations to achieve a given accuracy. Finally, we demonstrate the performance of the proposed algorithm and the theoretical findings for solving a randomly generated example of a distributed least squares problem.