  We use the duality between color and kinematics to obtain scattering
amplitudes in non-minimal conformal N=0,1,2,4 (super)gravity theories. Generic
tree amplitudes can be constructed from a double copy between
(super-)Yang-Mills theory and a new gauge theory built entirely out of
dimension-six operators. The latter theory is marginal in six dimensions and
contains modes with a wrong-sign propagator, echoing the behavior of conformal
gravity. The dimension-six Lagrangian is uniquely determined by demanding that
its scattering amplitudes obey the color-kinematics duality. The conformal
gravity amplitudes obtained from the double copy are compared with the
Berkovits-Witten twistor string and shown to agree up to at least eight points
in the MHV sector. Our construction can be generalized in a number of ways.
Adding scalars to the dimension-six theory gives Maxwell-Weyl gravity, and
further adding phi^3 self-interactions among these scalars gives
Yang-Mills-Weyl gravity. The latter is identified with Witten's twistor string
for maximal N=4 supersymmetry. Deforming the dimension-six theory by adding a
Yang-Mills term, m^2 F^2, gives a gauge theory that interpolates between
marginal D=6 and D=4 theories. The corresponding double copy gives an
interpolation between conformal gravity and Einstein gravity.
