We measure the static shear viscosity η in a two-component Fermi gas near a broad collisional (Feshbach) resonance, as a function of interaction strength and energy. We find that η has both a quadratic and a linear dependence on the interaction strength 1/(k(FI)a), where a is the s-wave scattering length and k(FI) is the Fermi wave vector for an ideal gas at the trap center. For energies above the superfluid transition, the minimum in η as a function of interaction strength is significantly shifted toward the BEC side of resonance, to 1/(k(FI)a)≃0.25.