Proposed quantum networks require both a quantum interface between light and matter and the coherent control of quantum states(1,2). A quantum interface can be realized by entangling the state of a single photon with the state of an atomic or solid-state quantum memory, as demonstrated in recent experiments with trapped ions(3,4), neutral atoms(5,6), atomic ensembles(7,8) and nitrogen-vacancy spins(9). The entangling interaction couples an initial quantum memory state to two possible light-matter states, and the atomic level structure of the memory determines the available coupling paths. In previous work, the transition parameters of these paths determined the phase and amplitude of the final entangled state, unless the memory was initially prepared in a superposition state(4) (a step that requires coherent control). Here we report fully tunable entanglement between a single Ca-40(+) ion and the polarization state of a single photon within an optical resonator. Our method, based on a bichromatic, cavity-mediated Raman transition, allows us to select two coupling paths and adjust their relative phase and amplitude. The cavity setting enables intrinsically deterministic, high-fidelity generation of any two-qubit entangled state. This approach is applicable to a broad range of candidate systems and thus is a promising method for distributing information within quantum networks.