A general method for computing excess chemical potentials is presented. The excess chemical potential of a solute or ligand molecule is estimated from the potential of mean-force (PMF) calculated along a nonphysical fourth spatial dimension, w, into which the molecule is gradually inserted or from which it is gradually abstracted. According to this "4D-PMF" (four dimensional) scheme, the free energy difference between two limiting states defines the excess chemical potential: At w = +/-infinity, the molecule is not interacting with the rest of the system, whereas at w=0, it is fully interacting. Use of a fourth dimension avoids the numerical instability in the equations of motion encountered upon growing or shrinking solute atoms in conventional free energy perturbation simulations performed in three dimensions, while benefiting from the efficient sampling of configurational space afforded by PMF calculations. The applicability and usefulness of the method are illustrated with calculations of the hydration free energy of simple Lennard-Jones (LJ) solutes, a water molecule, and camphor, using molecular dynamics simulations and umbrella sampling. Physical insight into the nature of the PMF profiles is gained from a continuum treatment of short- and long-range interactions. The short-range barrier for dissolution of a LJ solute in the added dimension provides an apparent surface tension of the solute. An approximation to the long-range behavior of the PMF profiles is made in terms of a continuum treatment of LJ dispersion and electrostatic interactions. Such an analysis saves the need for configurational sampling in the long-range limit of the fourth dimension. The 4D-PMF method of calculating excess chemical potentials should be useful for neutral solute and ligand molecules with a wide range of sizes, shapes, and polarities.