  We propose a novel hybrid quantum gate between an atom and a microwave photon
in a superconducting coplanar waveguide cavity by exploiting the strong
resonant microwave coupling between adjacent Rydberg states. Using
experimentally achievable parameters gate fidelities $> 0.99$ are possible on
sub-$\mu$s timescales for waveguide temperatures below 40 mK. This provides a
mechanism for generating entanglement between two disparate quantum systems and
represents an important step in the creation of a hybrid quantum interface
applicable for both quantum simulation and quantum information processing.
