We form single-electron transistors from individual chemically synthesized gold nanoparticles, 5-15 nm in diameter, with monolayers of organic molecules serving as tunnel barriers. These devices allow us to measure the discrete electronic energy levels of individual gold nanoparticles that are, by virtue of chemical synthesis, well-defined in their composition, size and shape. We show that the nanoparticles are nonmagnetic and have spectra in good accord with random-matrix-theory predictions taking into account strong spin-orbit coupling.