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If a small region of electron gas is separated from its leads by tunnel junctions, the quantization of charge and energy gives rise to sharp peaks in the conductance as a function of electron density, one for each electron added to the isolated region. Because the charge and energy are quantized, we call this an artificial atom. The peaks in conductance occur when the Fermi energy in the leads is degenerate with one of the quantized energy levels of the artificial atom, thus providing a measure of the chemical potential µNof the artificial atom with a fixed number Nof electrons. Detailed comparison of µNat high magnetic field with theory shows that exchange profoundly affects the ground state of electrons confined in this way. In particular, it stabilizes the singlet state to an unexpectedly high field, above which there is a gradual transition to the spin polarized state.