The Ward identities involving the currents associated to the spontaneously broken scale and special conformal transformations are derived and used to determine, through linear order in the two soft-dilaton momenta, the double-soft behavior of scattering amplitudes involving two soft dilatons and any number of other particles. It turns out that the double-soft behavior is equivalent to performing two single-soft limits one after the other. We confirm the new double-soft theorem perturbatively at tree-level in a D-dimensional conformal field theory model, as well as nonperturbatively by using the gravity dual of N = 4 super Yang-Mills on the Coulomb branch; i.e. the Dirac-Born-Infeld action on AdS(5) x S-5.