  We perform a lattice Monte-Carlo calculation of the trace-anomaly two-point
function at finite temperature in the SU(3) gauge theory. We obtain the
long-distance properties of the correlator in the continuum limit and extract
the bulk viscosity zeta via a Kubo formula. Unlike the tensor correlator
relevant to the shear viscosity, the scalar correlator depends strongly on
temperature. If s is the entropy density, we find that zeta/s becomes rapidly
small at high T, zeta/s<0.15 at 1.65T_c and zeta/s<0.015 at 3.2T_c. However
zeta/s rises dramatically just above T_c, with 0.5<zeta/s<2.0 at 1.02T_c.
