I will introduce a new double-copy construction for scattering amplitudes in conformal (super)gravity. The construction relies on input from two gauge theories: (super-)Yang-Mills and a new non-supersymmetric gauge theory that is marginal in six dimensions. The latter theory has several unusual features, such as negative norm states (as expected by its relation to conformal gravity) and a scalar in a peculiar representation of the gauge group. I will discuss various generalizations of the construction.