The non-Abelian analogue of electrical conductivity at high temperature has
previously been known only at leading logarithmic order - that is, neglect
ing effects suppressed only by an inverse logarithm of the gauge coupling.
We calculate the first sub-leading correction. This has immediate applicati
on to improving, to next-to-leading log order, both effective theories of n
on-perturbative color dynamics, and calculations of the hot electroweak bar
yon number violation rate.