We measure the shear viscosity in a two-component Fermi gas of atoms, tuned to a broad s-wave Feshbach resonance. At resonance, the atoms strongly interact and exhibit universal behavior, where the equilibrium thermodynamic properties and the transport coefficients are universal functions of the density $n$ and temperature $T$. We determine the shear viscosity in units of $\hbar\,n$ as a function of the reduced temperature at the trap center from two experiments, At low temperature, we use breathing mode damping to measure the shear viscosity. For high temperature, we use anisotropic expansion of the cloud to find viscosity, which exhibits the predicted $T^{3/2}$ scaling. The trap-averaged entropy per particle and shear viscosity are used to estimate the ratio of the shear viscosity to the entropy density, which is compared to that of a perfect fluid. The vanishing of bulk viscosity for a unitary Fermi gas will also be briefly discussed.