  We consider the double scaling limit in the random matrix ensemble with an
external source $\frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM$ defined on $n\times
n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm
a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues
of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one
interval for $0 < a < 1$. These two cases were treated in Parts I and II, where
we showed that the local eigenvalue correlations have the universal limiting
behavior known from unitary random matrix ensembles. For the critical case
$a=1$ new limiting behavior occurs which is described in terms of Pearcey
integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish
this result by applying the Deift/Zhou steepest descent method to a $3 \times
3$-matrix valued Riemann-Hilbert problem which involves the construction of a
local parametrix out of Pearcey integrals. We resolve the main technical issue
of matching the local Pearcey parametrix with a global outside parametrix by
modifying an underlying Riemann surface.
