  We consider 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 only two eigenvalues $\pm a$ of
equal multiplicity. For the case $a > 1$, we establish the universal behavior
of local eigenvalue correlations in the limit $n \to \infty$, which is known
from unitarily invariant random matrix models. Thus, local eigenvalue
correlations are expressed in terms of the sine kernel in the bulk and in terms
of the Airy kernel at the edge of the spectrum. We use a characterization of
the associated multiple Hermite polynomials by a $3 \times 3$-matrix
Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze
the Riemann-Hilbert problem in the large $n$ limit.
