The constant-number Monte Carlo method introduced by Matsoukas and co-workers for simulating particulate systems is applied to the kinetics of aggregating and fragmenting particles. The efficiency of this approach is increased by incorporating a modified version of Gillespie's full-conditioning algorithm for selecting an aggregation or fragmentation event. After the steps comprising the algorithm are outlined, it is validated by simulations for several aggregation and fragmentation kernels for which the population balance equations can be solved exactly. The results agree very well with the analytical expressions except for those kernels that give rise to a gelation transition, such as the product kernel k(ij)=ij. In this case, the simulation data are accurate below the transition time t(g), but deviate significantly above t(g). The accuracy of the simulation method in describing gelling kernels, including those of the form k(ij)=(ij)(omega), is also investigated. For a strongly gelling kernel, t(g) is accurately predicted by maxima in the time derivative of the second moment of the particle mass and the time dependence of the number of size classes in the simulation. Gel formation is simulated by setting a threshold size g above which particles have properties of the gel in the Stockmayer or Flory models. The Stockmayer model can be accurately simulated for a value of g that depends on the number of particles in the simulation. Simulation of the Flory model is less successful; results are obtained more efficiently by using the conventional constant-volume Monte Carlo method.