We study a single cluster of Lennard-Jones atoms using a novel and physically transparent Monte Carlo simulation technique. We compute the canonical ensemble averages of the grand canonical growth and decay probabilities of the cluster, and identify the critical cluster, the size for which the growth and decay probabilities are equal. The size and internal energy of the critical cluster, for different values of the temperature and chemical potential, are used together with the nucleation theorems to predict the behavior of the nucleation rate as a function of these parameters. Our results agree with those found in the literature, and roughly correspond to the predictions of classical theory. In contrast to most other simulation studies, we are able to concentrate on the properties of the clusters which are most important to the process of nucleation, namely those around the critical size. This makes our simulations computationally more efficient. (C) 2000 American Institute of Physics. [S0021- 9606(00)50209-8].