  We propose a superspace formulation for the Weyl multiplet of N=1 conformal
supergravity in five dimensions. The corresponding superspace constraints are
invariant under super-Weyl transformations generated by a real scalar
parameter. The minimal supergravity multiplet, which was introduced by Howe in
1981, emerges if one couples the Weyl multiplet to an Abelian vector multiplet
and then breaks the super-Weyl invariance by imposing the gauge condition W=1,
with W the field strength of the vector multiplet. The geometry of superspace
is shown to allow the existence of a large family of off-shell supermultiplets
that possess uniquely determined super-Weyl transformation laws and can be used
to describe supersymmetric matter. Many of these supermultiplets have not
appeared within the superconformal tensor calculus. We formulate a manifestly
locally supersymmetric and super-Weyl invariant action principle. In the
super-Weyl gauge W=1, this action reduces to that constructed in
arXiv:0712.3102. We also present a superspace formulation for the dilaton Weyl
multiplet.
