  We compute the two-point angular correlation function $w(\theta)$ for a
sample of $\sim$ 1700 galaxies to a magnitude-limit equivalent to $R \sim 29.5$
using a catalog derived from the Hubble Deep Field images. A non zero value of
$w(\theta)$ is measured down to $R=29.0$. The amplitude of $w(\theta)$ at the
bright magnitude limit ($R \sim 26$) is consistent with previous ground-based
observations. At fainter magnitudes the clustering amplitude continues to
decrease but at a slower rate than that predicted by the power law
$w(1'')\propto 10^{-0.27R}$ observed for shallower samples. The observed
$w(\theta)$ over the magnitude range $20 <R< 29$ is consistent with linear
evolution of the clustering of a galaxy population which at present has a
correlation length $r_0$ of about 4 h^(-1) Mpc, close to that of local IRAS
galaxies. We also investigate the impact that magnification bias induced by
weak gravitational lensing may have on our results. Although the observed
amplitude of $w(\theta)$ can differ from the true amplitude by up to 30%, this
effect is not large enough to affect our conclusions. Finally, by using a
color-selected sample, we examine whether the expected effects of magnification
bias can be used for an independent determination of cosmological parameters in
deep images. We conclude that the amplitude of the effect can be large and in
some cases even produce an upturn of the amplitude of the correlation with
limiting magnitude. However, we find that it is not possible to detect the
effects of magnification bias on $w(\theta)$ from images alone. If redshift
information becomes available, it is possible to measure the effects of
magnification bias directly and thus constrain the density parameter $\Omega_0$
and the bias factor b.
