  We give an analytic demonstration that the 3+1 dimensional large N SU(N) pure
Yang-Mills theory, compactified on a small 3-sphere so that the coupling
constant at the compactification scale is very small, has a first order
deconfinement transition as a function of temperature. We do this by explicitly
computing the relevant terms in the canonical partition function up to 3-loop
order; this is necessary because the leading (1-loop) result for the phase
transition is precisely on the borderline between a first order and a second
order transition. Since numerical work strongly suggests that the infinite
volume large N theory also has a first order deconfinement transition, we
conjecture that the phase structure is independent of the size of the 3-sphere.
To deal with divergences in our calculations, we are led to introduce a novel
method of regularization useful for nonabelian gauge theory on a 3-sphere.
