We propose a scheme for teleporting an arbitrary superposition of entangled Dicke states of any number of atoms squbitsd between two distant cavities. Our method relies on adiabatic passage using multi-atom dark states in each cavity, and a conditional detection of photons leaking out of both cavities. The ideal success probability of the protocol decreases polynomially in the number of atoms. The fidelity is unity for a single Dicke state, and can be optimized for the superposition by unitary postprocessing. Issues of experimental feasibility and applications to quantum informatics are discussed.