  In this paper we discuss the blackhole-string transition of the small
Schwarzschild blackhole of $AdS_5 \times S^5$ using the AdS/CFT correspondence
at finite temperature. The finite temperature gauge theory effective action, at
weak {\it and} strong coupling, can be expressed entirely in terms of constant
Polyakov lines which are $SU (N)$ matrices. In showing this we have taken into
account that there are no Nambu-Goldstone modes associated with the fact that
the 10 dimensional blackhole solution sits at a point in $S^5$. We show that
the phase of the gauge theory in which the eigenvalue spectrum has a gap
corresponds to supergravity saddle points in the bulk theory. We identify the
third order $N = \infty$ phase transition with the blackhole-string transition.
This singularity can be resolved using a double scaling limit in the transition
region where the large N expansion is organized in terms of powers of
$N^{-2/3}$. The $N = \infty$ transition now becomes a smooth crossover in terms
of a renormalized string coupling constant, reflecting the physics of large but
finite N. Multiply wound Polyakov lines condense in the crossover region. We
also discuss the implications of our results for the resolution of the
singularity of the Lorenztian section of the small Schwarzschild blackhole.
