Distributed quantum computing requires rapid, high-fidelity entanglement between qubits in separate processor modules. Photonic links between single-atom qubits provide switchable, long-range connections, but photon loss limits their entanglement rate. Here we entangle a trapped ¹³⁸Ba⁺ ion with a silicon-vacancy (SiV⁻) center in a diamond nanophotonic cavity. A photon emitted by the ion is converted in wavelength and encoding, reflected from the SiV⁻--cavity system and detected, so that a single detected photon heralds entanglement. Including all conversion losses, we generate ion–SiV⁻ Bell pairs at an average rate of 1.03(1) kHz with a fidelity of 87.9(7)%. This is four times the rate of the fastest photonic link between two trapped ions and corresponds to one Bell pair per millisecond, matching the cycle time of planned trapped-ion processors.