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.

24 pages, 8 figures