  We show that the Kontsevich integral on $n\times n$ matrices ($n< \infty$) is
the isomonodromic tau function associated to a $2\times 2$ Riemann--Hilbert
problem. The approach allows us to gain control of the analysis of the
convergence as $n\to\infty$. By an appropriate choice of the external source
matrix in Kontsevich's integral, we show that the limit produces the
isomonodromic tau function of a special tronqu\'ee solution of the first
Painlev\'e hierarchy, and we identify the solution in terms of the Stokes' data
of the associated linear problem. We also show that there are several tau
functions that are analytic in appropriate sectors of the space of parameters
and that the formal Witten-Kontsevich tau function is the asymptotic expansion
of each of them in their respective sectors, thus providing an analytic tool to
analyze its nonlinear Stokes' phenomenon.
