In the Bargmann representation of quantum mechanics, physical states are mapped into entire functions of a complex variable z*, whereas the creation and annihilation operators (a) over cap (dagger) and (a) over cap a play the role of multiplication and differentiation with respect to z*, respectively. In this paper we propose an alternative representation of quantum states, conjugate to the Bargmann representation, where the roles of (a) over cap (dagger) and (a) over cap a are reversed, much like the roles of the position and momentum operators in their respective representations. We derive expressions for the inner product that maintain the usual notion of distance between states in the Hilbert space. Applications to simple systems and to the calculation of semiclassical propagators are presented.4210