  We report the observation of spin segregation, i.e., separation of spin
density profiles, in a trapped ultracold Fermi gas of $^6$Li with a
magnetically tunable scattering length close to zero. The roles of the spin
components are inverted when the sign of the scattering length is reversed. The
observed density profiles are in qualitative agreement with the spin-wave
theory applied previously to explain spin segregation in a Bose gas, but
disagree in amplitude by two orders of magnitude. The observed atomic density
profiles are far from equilibrium, yet they persist for $\simeq$ 5 seconds in a
trap with an axial frequency of $\simeq$ 150 Hz. We attribute this long
lifetime to Fermi statistics: The scattering amplitude is nonzero only for
atoms in opposite states, and vanishes for atoms in the same state. By
measuring the magnetic field at which spin segregation ceases, we precisely
determine the zero crossing in the scattering length of $^6$Li as $527.5\pm0.2$
G.
