  We consider an abelian N=4 super Yang-Mills theory coupled to background N=4
conformal supergravity fields. At the classical level, this coupling is
invariant under global SU(1,1) transformation of the complex ("dilaton-axion")
supergravity scalar combined with an on-shell vector-vector duality. We compute
the divergent part of the corresponding quantum effective action found by
integrating over the super Yang-Mills fields and demonstrate its SU(1,1)
invariance. This divergent part related to the conformal anomaly is one-loop
exact and should be given by the N=4 conformal supergravity action containing
the Weyl tensor squared term. This allows us to determine the full non-linear
form of the bosonic part of the N=4 conformal supergravity action which has
manifest SU(1,1) invariance.
