As a double generalization of biFrobenius algebras and weak Hopf algebra, we introduce and study weak biFrobenius algebras (or briefly wbF algebras). A wbF algebra is a Frobenius algebra and a Frobenius coalgebra and satisfies some compatibility conditions. We discuss more properties of wbF algebras keeping as close as possible to the classical theory of weak Hopf algebras, and give conditions for finite dimensional algebras and coalgebras to be wbF algebras. Then we studies substructures in wbF algebras. Finally, we prove that our wbF algebras can give rise to Frobenius double algebras in sense of Szlachányi.