  An effective field theory approach is developed for calculating the
thermodynamic properties of a field theory at high temperature $T$ and weak
coupling $g$. The effective theory is the 3-dimensional field theory obtained
by dimensional reduction to the bosonic zero-frequency modes. The parameters of
the effective theory can be calculated as perturbation series in the running
coupling constant $g^2(T)$. The free energy is separated into the contributions
from the momentum scales $T$ and $gT$, respectively. The first term can be
written as a perturbation series in $g^2(T)$. If all forces are screened at the
scale $gT$, the second term can be calculated as a perturbation series in
$g(T)$ beginning at order $g^3$. The parameters of the effective theory satisfy
renormalization group equations that can be used to sum up leading logarithms
of $T/(gT)$. We apply this method to a massless scalar field with a $\Phi^4$
interaction, calculating the free energy to order $g^6 \log g$ and the
screening mass to order $g^5 \log g$.
