  We solve the 3-loop $\Phi$-derivable approximation to the thermodynamics of
the massless $\phi^4$ field theory by reducing it to a 1-parameter variational
problem. The thermodynamic potential is expanded in powers of $g^2$ and $m/T$,
where $g$ is the coupling constant, $m$ is a variational mass parameter, and
$T$ is the temperature. There are ultraviolet divergences beginning at 6th
order in $g$ that cannot be removed by renormalization. However the finite
thermodynamic potential obtained by truncating after terms of 5th order in $g$
and $m/T$ defines a stable approximation to the thermodynamic functions.
