We report the observation of spin segregation, i.e., time-dependent separation of the spin density profiles of two spin states, in a trapped, coherently prepared Fermi gas of 6Li with a magnetically tunable scattering length a12 close to zero. For |a12| approximately = 5 bohr, as the cloud profiles evolve, the measured difference in the densities at the cloud center increases in 200 ms from 0 to approximately = 60% of the initial mean density and changes sign with a12. The data are in disagreement in both amplitude and temporal evolution with a spin-wave theory for a Fermi gas. In contrast, for a Bose gas, an analogous theory has successfully described previous observations of spin segregation. The observed segregated atomic density profiles are far from equilibrium, yet they persist for approximately = 5 s, long compared to the axial trapping period of 6.9 ms. We find the zero crossing in a12=0, where spin segregation ceases, at 527.5+/-0.2 G.