  The free energy for QCD at high temperature $T$ is calculated to order $g^5$
using effective-field-theory methods to separate the contributions from the
momentum scales $T$ and $gT$. The effects of the scale $T$ enter through the
coefficients in the effective lagrangian for the 3-dimensional effective theory
obtained by dimensional reduction. The perturbation series for these
coefficients seem to be well-behaved if the running coupling constant is
sufficiently small: $\alpha_s(2 \pi T) \ll 1$. For the contribution to the free
energy from the scale $gT$, the perturbation series is well-behaved only if
$\alpha_s(2 \pi T)$ is an order of magnitude smaller. The implication for
applications of perturbative QCD to the quark-gluon plasma are briefly
discussed.
