A method is described for calculating from first principles the coefficients in the Becker-D/Jring equations for the rate of change of the distribution of cluster sizes in a low-density lattice gas with Kawasaki dynamics. The method depends on solving a diffusion problem for the concentration of particles near a given cluster. The coefficients are calculated for cluster sizes up to 6, on a simple cubic lattice at a temperature 0.59 times the critical temperatures, and extrapolated to larger sizes. The resulting version of the Becker-D~ring equations is then solved numerically. Comparison with the results of a computer simulation (at overall
concentration 0.075) carried out by Kalos and others indicates that the method gives quite good predictions of the dependence of the cluster distribution on the critical cluster size (usually denoted by l*) but that the predicted rate of change of critical cluster size with time is too small, at this overall concentration, by a factor of about 0.3.