  We investigate the matrix model with weight $w(x):=\exp(-z^2/2x^2 + t/x -
x^2/2)$ and unitary symmetry. and unitary symmetry. In particular we study the
double scaling limit as $N \to \infty$ and $(\sqrt{N} t, Nz^2 ) \to (u_1,u_2)$,
where $N$ is the matrix dimension and the parameters $(u_1,u_2)$ remain finite.
Using the Deift-Zhou steepest descent method we compute the asymptotics of the
partition function when $z$ and $t$ are of order $O\bigl(N^{-1/2}\bigr)$. In
this regime we discover a phase transition in the $(z,N)$-plane characterised
by the Painleve' III equation. This is the first time that Painleve' III
appears in studies of double scaling limits in Random Matrix Theory and is
associated to the emergence of an essential singularity in the weighting
function. The asymptotics of the partition function is expressed in terms of a
particular solution of the Painleve' III equation. We derive explicitly the
initial conditions in the limit $Nz^2\rightarrow u_2$ of this solution.
