We present a simple method to estimate the solubility of weakly soluble solids in different solvents. The method involves the calculation of the residual chemical potential of a single solute molecule in the solvent of interest using an appropriate atomistic free energy simulation technique. In the present method, an expanded ensemble calculation is used, along with a combined Wang-Landau/Bennett's acceptance ratio method. To avoid the simulation of the solid phase and the use of analytic reference states, a single experimental solubility data point for the solute in a single reference solvent is also required. The method has advantages over more empirical descriptor-based methods in that the simulations enable insight into the underlying molecular driving forces responsible for solubility trends. Results are presented for the solubility of 1,2,4,5-tetramethylbenzene, phenanthrene, and fluorene in the solvents hexane, octane, ethanol, 1-octanol, cyclohexane, benzene, and 1,4-dioxane. Overall, agreement between the results of the present study and available experimental data is good. In general, the predictions have a lower total absolute error when compared to experimental solubility data than those obtained using the Abraham general solvation model.