  We calculate the shear viscosity $\eta$ in the quark-gluon plasma (QGP) phase
within a virial expansion approach with particular interest in the ratio of
$\eta$ to the entropy density $s$, i.e. $\eta/s$. The virial expansion approach
allows us to include the interactions between the partons in the deconfined
phase and to evaluate the corrections to a single-particle partition function.
In the latter approach we start with an effective interaction with parameters
fixed to reproduce thermodynamical quantities of QCD such as energy and/or
entropy density. We also directly extract the effective coupling $\ga_{\rm V}$
for the determination of $\eta$. Our numerical results give a ratio
$\eta/s\approx 0.097$ at the critical temperature $T_{\rm c}$, which is very
close to the theoretical bound of $1/(4\pi)$. Furthermore, for temperatures
$T\leq 1.8 T_{\rm c}$ the ratio $\eta/s$ is in the range of the present
experimental estimates $0.1-0.3$ at RHIC. When combining our results for
$\eta/s$ in the deconfined phase with those from chiral perturbation theory or
the resonance gas model in the confined phase we observe a pronounced minimum
of $\eta/s$ close to the critical temperature $T_{\rm c}$.
