We follow the classical double copy (DC) procedure that links Yang-Mills and double field theory (DFT), and apply it on a four-derivative gauge theory which is known to be related to Weyl gravity at the level of the amplitudes. We obtain a perturbative T-duality invariant theory on a double geometry, or conformal double field theory (CDFT), incorporating Weyl gravity plus $b$-field and dilaton contributions at quadratic order, without the need to impose a gauge-fixing condition. We also extend the formulation to cubic order for the case of vanishing generalized dilaton, which still incorporates Weyl gravity when $\square h_{\mu \nu }=h=0$. CDFT, together with ordinary DFT, are examples of T-duality invariant theories constructed through classical DC maps, revealing a promising and deep connection between gauge theories and T-duality invariant models.