  We calculate the negative integer moments of the (regularized) characteristic
polynomials of N x N random matrices taken from the Gaussian Orthogonal
Ensemble (GOE) in the limit as $N \to \infty$. The results agree nontrivially
with a recent conjecture of Berry & Keating motivated by techniques developed
in the theory of singularity-dominated strong fluctuations. This is the first
example where nontrivial predictions obtained using these techniques have been
proved.
