  Rota-Baxter operators and bialgebras go hand in hand in their applications,
such as in the Connes-Kreimer approach to renormalization and the operator
approach to the classical Yang-Baxter equation. We establish a bialgebra
structure that is compatible with the Rota-Baxter operator, called the
Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is
characterized by generalizations of matched pairs of algebras and double
constructions of Frobenius algebras to the context of Rota-Baxter algebras. The
study of the coboundary case leads to an enrichment of the associative
Yang-Baxter equation (AYBE) to Rota-Baxter algebras. Antisymmetric solutions of
the equation are used to construct Rota-Baxter ASI bialgebras. The notions of
an $\mathcal{O}$-operator on a Rota-Baxter algebra and a Rota-Baxter dendriform
algebra are also introduced to produce solutions of the AYBE in Rota-Baxter
algebras and thus to provide Rota-Baxter ASI bialgebras. An unexpected
byproduct is that a Rota-Baxter ASI bialgebra of weight zero gives rise to a
quadri-bialgebra instead of bialgebra constructions for the dendriform algebra.
