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 [Formula Presented] atoms in an applied magnetic field near a Feshbach resonance, where strong interactions are observed. We find that 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. © 2003 The American Physical Society.