We present in this paper the application of the real-space dynamic renormalization-group method to a two-dimensional kinetic Ising model with a conserved order parameter. We obtain the diffusion coefficient as a function of the temperature in the conventional approximation. We show that our renormalization procedure automatically preserves the conservation laws upon iteration. We present recursion relations and results for the time-dependent correlation functions, with particular emphasis put on the evaluation of those dynamic correlations which are obtainable in field-emission experiments. The proper hydrodynamic behavior is obtained at long times.