An evaporative cooling strategy that uses a two-component Fermi gas was employed to cool a magnetically trapped gas of 7 X 10(5) K-40 atoms to 0.5 of the Fermi temperature T-F. In this temperature regime, where the state occupation at the Lowest energies has increased from essentially zero at high temperatures to nearly 60 percent, quantum degeneracy was observed as a barrier to evaporative cooling and as a modification of the thermodynamics. Measurements of the momentum distribution and the total energy of the confined Fermi gas directly revealed the quantum statistics.