Recent experiments on the shear viscosity η in a unitary Fermi gas fail to see the theoretically predicted upturn in η at the lower T. In this Letter, we compute η in a fashion which is demonstrably consistent with conservation laws and, in the process, provide an understanding of recent experiments. We show that this disagreement with prior theories cannot be readily attributed to the trap, since (via edge effects) trap-averaged viscosities will be larger than their homogeneous counterparts. The small values of η we find can be simply understood; they reflect the fact that the Goldstone bosons (phonons) do not couple to transverse probes such as η, and fermionic excitations, which determine the viscosity, are necessarily absent in the ground state.