  In this paper, we construct a bialgebra theory for associative conformal
algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one
hand, it is an attempt to give conformal structures for antisymmetric
infinitesimal bialgebras. On the other hand, under certain conditions, such
structures are equivalent to double constructions of Frobenius conformal
algebras, which are associative conformal algebras that are decomposed into the
direct sum of another associative conformal algebra and its conformal dual as
$\mathbb{C}[\partial]$-modules such that both of them are subalgebras and the
natural conformal bilinear form is invariant. The coboundary case leads to the
introduction of associative conformal Yang-Baxter equation whose antisymmetric
solutions give antisymmetric infinitesimal conformal bialgebras. Moreover, the
construction of antisymmetric solutions of associative conformal Yang-Baxter
equation is given from $\mathcal{O}$-operators of associative conformal
algebras as well as dendriform conformal algebras.
