We construct a variational explicit-solute implicit-solvent model for the solvation of molecules. Central in this model is an effective solvation free-energy functional that depends solely on the position of solute-solvent interface and solute atoms. The total free energy couples altogether the volume and interface energies of solutes, the solute-solvent van der Waals interactions, and the solute-solute mechanical interactions. A curvature dependent surface tension is incorporated through the so-called Tolman length which serves as the only fitting parameter in the model. Our approach extends the original variational implicit-solvent model of Dzubiella, Swanson, and McCammon [Phys. Rev. Lett. 2006, 96, 087802 and J. Chem. Phys. 2006, 124, 084905] to include the solute molecular mechanics. We also develop a novel computational method that combines the level-set technique with optimization algorithms to determine numerically the equilibrium conformation of nonpolar molecules. Numerical results demonstrate that our new model and methods can capture essential properties of nonpolar molecules and their interactions with the solvent. In particular, with a suitable choice of the Tolman length for the curvature correction to the surface tension, we obtain the solvation free energy for a benzene molecule in a good agreement with experimental results.