  We compute the bulk viscosity zeta of high-temperature QCD to leading order
in powers of the running coupling alpha_s(T). We find that it is negligible
compared to shear viscosity eta for any alpha_s that might reasonably be
considered small. The physics of bulk viscosity in QCD is a bit different than
in scalar phi^4 theory. In particular, unlike in scalar theory, we find that an
old, crude estimate of zeta as 15 ((1/3)-v_s^2)^2 eta gives the correct order
of magnitude, where v_s is the speed of sound. We also find that leading-log
expansions of our result for zeta are not accurate except at very small
coupling.
