  The conductance resulting from resonant tunneling through a droplet of $N
\sim 30$ electrons is used to measure its chemical potential $\mu_N$. Abrupt
shifts of $\mu_N$ occur at sharply defined values of the magnetic field, at
which the state of the droplet changes. These are used to study part of the
phase-diagram of the droplet in strong magnetic fields; we find evidence for a
new phase in the spin polarized regime. We make a detailed comparison between
theory and experiment: Hartree-Fock provides a quantitative description of the
measurements when both spin-split states of the lowest orbital Landau level are
occupied and a qualitative one in the spin polarized regime.
