We study the Kondo effect in a vertical quantum dot which is strongly coupled to leads. The electron number, N, and spin state, S, are clearly defined in this "artificial atom". We observe the Kondo effect in the low temperature transport characteristics when the dot has only one electron (N=l) and spin S=1/2. Furthermore, we observe a much stronger Kondo effect for N=2 when the spin singlet (S=0) - triplet (S=1) degeneracy is induced by magnetic field. We investigate the crossover of the Kondo effect when the two-electron ground state is changed from a spin-singlet via a singlet-triplet (S-T) degeneracy to a spin-triplet by sweeping the magnetic field. No Kondo effect is observed for the singlet ground state. The Kondo temperature is highest for the S-T degeneracy, and gradually decreases as the degeneracy is lifted such that the triplet is only the ground state. This behavior is consistent with the recent theory.