The mechanism of "burst nucleation" of crystals from a supersaturated solution, as first put forward by LaMer and Dinegar (J. Amer. Chem. Soc. 72, 4847 (1950)), has been widely used in explaining the chemical synthesis of both mono- and poly-crystalline colloidal particles from solution. However, to our knowledge, the proposed kinetics of the burst nucleation mechanism have not been fully verified. Here we report the results of numerical simulations of a model of the burst nucleation process, in which instant rethermalization of sub-critical embryos and irreversible diffusive growth of super-critical clusters is assumed. We present approximate analytical results for the long-time behavior of the cluster-size distribution in the model, and discuss the applicability of the model and numerical results to experimental colloidal systems.