  We study conformal gravity as an alternative theory of gravitation. For
conformal gravity to be phenomenologically viable requires that the conformal
symmetry is not manifest at the energy scales of the other known physical
forces. Hence we require a mechanism for the spontaneous breaking of conformal
invariance. In this paper we study the possibility that conformal invariance is
spontaneously broken due to interactions with conformally coupled matter
fields. The vacuum of the theory admits conformally non-invariant solutions
corresponding to maximally symmetric space-times and variants thereof. These
are either de Sitter space-time or anti-de Sitter space-time in the full four
space-time dimensions or in a lower dimensional sub-space. We consider in
particular normalizable, linearized gravitational perturbations around the
anti-de Sitter background. Exploiting the conformal flatness of this
space-time, we show to second order, that these gravitational fluctuations,
that are taken to be fourier decomposable, carry zero energy-momentum. This
squares well with the theorem that asymptotically flat space-times conformal
gravity contain zero energy and momentum \cite{bhs}. We also show the
possibility of domain wall solitons interpolating between the ground states of
spontaneously broken conformal symmetry that we have found. These solitons
necessarily require the vanishing of the scalar field, repudiating the recent
suggestion \cite{f} that the conformal symmetry could be quarantined to a
sterile sector of the theory by choosing an appropriate field redefinition.
