We compute the shear viscosity of a superfluid atomic Fermi gas in the unitarity limit. The unitarity limit is characterized by a divergent scattering length between the atoms, and it has been argued that this will result in a very small viscosity. We show that in the low temperature
T limit the shear viscosity scales as ${\ensuremath{\xi}}^{5}∕{T}^{5}$, where the universal parameter $\ensuremath{\xi}$ relates the chemical potential and the Fermi energy, $\ensuremath{\mu}=\ensuremath{\xi}{\ensuremath{\epsilon}}_{F}$. Combined with the high temperature expansions of the viscosity our results suggest that the viscosity has a minimum near the critical temperature
Tc. A na\"{\i}ve extrapolation indicates that the minimum value of the ratio of viscosity over entropy density is within a factor of $\ensuremath{\sim}5$ of the proposed bound $\ensuremath{\eta}∕s\ensuremath{\ge}\ensuremath{\hbar}∕(4\ensuremath{\pi}{k}_{B})$.