  Direct photon production in longitudinally polarised proton-proton collisions
offers the most direct and unproblematic possibility to determine the polarised
gluon distribution of a proton. This information could play a major role for
improving our understanding of the nucleon structure and QCD in general. It is
hoped that such experiments will be done at RHIC. We present results of
detailed Monte Carlo simulations using a code called {\sc SPHINX}. We find that
for RHIC energies and large gluon polarisation the Compton graph dominates
allowing for a direct test of $\Delta g$. Triggering on away-side jets with the
envisaged jet-criteria should allow to obtain more detailed information on
$\Delta g(x)$. The photon asymmetry resulting from the asymmetry of produced
$\pi^0$'s provides an additional signal, which is complementary to the other
two. For small gluon polarisation, i.e. $\Delta g \le 0.5$ or very soft
polarised gluon-distributions the envisaged experiments will require a highly
sophisticated simulation and large statistics to extract more than upper bounds
for $|\Delta g(x)|$.}
