  We determine the shear viscosity of the ultracold Fermi gas at unitarity in
the normal phase using hydrodynamic expansion data. The analysis is based on a
generalized fluid dynamic framework which ensures a smooth transition between
the fluid dynamic core of the cloud and the ballistic corona. We use expansion
data taken by Joseph et al. and measurements of the equation of state by Ku et
al. We find that the shear viscosity to particle density ratio just above the
critical temperature is $\eta/n|_{T_c}=0.41\pm 0.11$. We also obtain evidence
that the shear viscosity to entropy density ratio has a minimum slightly above
$T_c$ with $\eta/s |_{\it min}=0.50\pm 0.10$.
