  A strongly-attractive, two-component Fermi gas of atoms exhibits universal
behavior and should be mechanically stable as a consequence of the quantum
mechanical requirement of unitarity. This requirement limits the maximum
attractive force to a value smaller than that of the outward Fermi pressure. To
experimentally demonstrate this stability, we use all-optical methods to
produce a highly degenerate, two-component gas of $^6$Li atoms in an applied
magnetic field near a Feshbach resonance, where strong interactions are
observed. We find that the gas is stable at densities far exceeding that
predicted previously for the onset of mechanical instability. Further, we
provide a temperature-corrected measurement of an important, universal,
many-body parameter which determines the stability--the mean field contribution
to the chemical potential in units of the local Fermi energy.
