  Ultrarelativistic heavy-ion collisions at the Relativistic Heavy-Ion Collider
(RHIC) are thought to have produced a state of matter called the
Quark-Gluon-Plasma, characterized by a very small shear viscosity to entropy
density ratio $\eta/s$, near the lower bound predicted for that quantity by
Anti-deSitter space/Conformal Field Theory (AdS/CFT) methods. As the produced
matter expands and cools, it evolves through a phase described by a hadron gas
with rapidly increasing $\eta/s$. We calculate $\eta/s$ as a function of
temperature in this phase and find that its value poses a challenge for viscous
relativistic hydrodynamics, which requires small values of $\eta/s$ throughout
the entire evolution of the reaction in order to successfully describe the
collective flow observables at RHIC. We show that the inclusion of non-unit
fugacities will reduce $\eta/s$ in the hadronic phase, yet not sufficiently to
be compatible with viscous hydrodynamics. We therefore conclude that the origin
of the low viscosity matter at RHIC must be in the partonic phase of the
reaction.
