  A bordering of GUE matrices is considered, in which the bordered row consists
of zero mean complex Gaussians N$[0,\sigma/2] + i {\rm N}[0,\sigma/2]$ off the
diagonal, and the real Gaussian N$[\mu,\sigma/\sqrt{2}]$ on the diagonal. We
compute the explicit form of the eigenvalue probability function for such
matrices, as well as that for matrices obtained by repeating the bordering. The
correlations are in general determinantal, and in the single bordering case the
explicit form of the correlation kernel is computed. In the large $N$ limit it
is shown that $\mu$ and/or $\sigma$ can be tuned to induce a separation of the
largest eigenvalue. This effect is shown to be controlled by a single
parameter, universal correlation kernel.
