  The diffractive process $\gamma p\to V + X$ (where $V$ is a vector meson and
$X$ results from the dissociation of the proton) is studied. In particular, we
consider the region of large momentum transfer (i.e. $|t|\gg\Lambda_{QCD}^2$)
and large centre-of-mass (CM) energy, $s$. The asymptotic $(s \to \infty,
s/|t|\gg 1$) behaviour is derived from the BFKL equation and compared to that
which is obtained in the Born approximation (two-gluon exchange). We also
calculate the corrections to the Born graphs by iterating the BFKL kernel
numerically. Improved convergence of the BFKL series is found by summing the
logarithms which occur when an exchanged gluon goes nearly on shell.
Importantly, we find evidence that the asymptotic solution to the BFKL equation
is inappropriate over most of the HERA range and we provide more realistic
predictions for the cross section. The predicted cross section is not too small
and can be measured at HERA, up to momentum transfers $|t| \sim 10$GeV$^2$.
