  We compute the leading and next--to--leading corrections to the finite
temperature scalar potential for a 3+1 dimensional $\phi^4$ theory using a
systematic $1/N$ expansion. Our approach automatically avoids problems
associated with infrared divergences in ordinary perturbation theory in
$\hbar$. The leading order result does not admit a first order phase
transition. The subleading result shows that the exact theory can admit at best
only a very weak first order phase transition. For $N=4$ and weak scalar
coupling we find that $T_1$, the temperature at which tunneling from the origin
may begin in the case of a first order transition, must be less than about 0.5
percent larger than $T_2$, the temperature at which the origin changes from
being a local minimum to being a local maximum. We compare our results to the
effective potential found from a sum of daisy graphs.
