We propose a 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 textgreater0.99 are possible on submicrosecond time scales 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.