  The free energy of a nonabelian gauge theory at high temperature $T$ can be
calculated to order $g^5$ using resummed perturbation theory, but the method
breaks down at order $g^6$. A new method is developed for calculating the free
energy to arbitrarily high accuracy in the high temperature limit. It involves
the construction of a sequence of two effective field theories by first
integrating out the momentum scale $T$ and then integrating out the momentum
scale $g T$. The free energy decomposes into the sum of three terms,
corresponding to the momentum scales $T$, $gT$, and $g^2T$. The first term can
be calculated as a perturbation series in $g^2(T)$, where $g(T)$ is the running
coupling constant. The second term in the free energy can be calculated as a
perturbation series in $g(T)$, beginning at order $g^3$. The third term can
also be expressed as a series in $g(T)$ beginning at order $g^6$, with
coefficients that can be calculated using lattice simulations of 3-dimensional
QCD. Leading logarithms of $T/(gT)$ and of $gT/(g^2T)$ can be summed up using
renormalization group equations.
