  The temperature dependence of conductance resonances is used to measure the
evolution with the magnetic field of the average level spacing $\Delta\epsilon$
of a droplet containing $\sim 30$ electrons created by lateral confinement of a
two-dimensional electron gas in GaAs. $\Delta\epsilon$ becomes very small ($<
30\mu$eV) near two critical magnetic fields at which the symmetry of the
droplet changes and these decreases of $\Delta\epsilon$ are predicted by
Hartree-Fock (HF) for charge excitations. Between the two critical fields,
however, the largest measured $\Delta\epsilon= 100\mu$eV is an order of
magnitude smaller than predicted by HF but comparable to the Zeeman splitting
at this field, which suggests that the spin degrees of freedom are important.
PACS: 73.20.Dx, 73.20.Mf
