We present a way to transfer quantum information of a photonic cat-state qubit (cqubit) between two cavities coupled to an artificial atom. The two logic states of the photonic cqubit here are encoded by two orthogonal cat states (i.e. superpositions of coherent states) of photons. We show that quantum information of one photonic cqubit can be deterministically transferred between two cavities, with a single-step operation only. As a specific example, we numerically analyse the experimental implementation of this proposal, by considering a set-up of two microwave cavities coupled to a superconducting artificial atom. This proposal can also be applied to transfer one-cqubit quantum information between two modes of a quantum field coupled by a natural or artificial atom.