  We study conformally-invariant theories of gravity in six dimensions. In four
dimensions, there is a unique such theory that is polynomial in the curvature
and its derivatives, namely Weyl-squared, and furthermore all solutions of
Einstein gravity are also solutions of the conformal theory. By contrast, in
six dimensions there are three independent conformally-invariant polynomial
terms one could consider. There is a unique linear combination (up to overall
scale) for which Einstein metrics are also solutions, and this specific theory
forms the focus of our attention in this paper. We reduce the equations of
motion for the most general spherically-symmetric black hole to a single
5th-order differential equation. We obtain the general solution in the form of
an infinite series, characterised by 5 independent parameters, and we show how
a finite 3-parameter truncation reduces to the already known Schwarzschild-AdS
metric and its conformal scaling. We derive general results for the
thermodynamics and the first law for the full 5-parameter solutions. We also
investigate solutions in extended theories coupled to conformally-invariant
matter, and in addition we derive some general results for conserved charges in
cubic-curvature theories in arbitrary dimensions.
