  We construct Gibbs perturbations of the Gamma process on $\mathbbm{R}^d$,
which may be used in applications to model systems of densely distributed
particles. First we propose a definition of Gibbs measures over the cone of
discrete Radon measures on $\mathbbm{R}^d$ and then analyze conditions for
their existence. Our approach works also for general L\'evy processes instead
of Gamma measures. To this end, we need only the assumption that the first two
moments of the involved L\'evy intensity measures are finite. Also uniform
moment estimates for the Gibbs distributions are obtained, which are essential
for the construction of related diffusions. Moreover, we prove a Mecke type
characterization for the Gamma measures on the cone and an FKG inequality for
them.
