Trapped ions are promising candidates for nodes of a scalable quantum network due to their long-lived qubit coherence times and high-fidelity single-and two-qubit gates. Future quantum networks based on trapped ions will require a scalable way to route photons between different nodes. Photonic integrated circuits from fabrication foundries provide a compact solution to this problem. However, these circuits typically operate at telecommunication wavelengths that are incompatible with the strong dipole emissions of trapped ions. In this work, we demonstrate the routing of single photons from a trapped ion using a photonic integrated circuit. We employ quantum frequency conversion to match the emission of the ion to the operating wavelength of a foundry-fabricated silicon nitride photonic integrated circuit, achieving a total transmission of 31.0% +/- 0.9% through the device. Using programmable phase shifters, we switch the single photons between the output channels of the circuit and demonstrate a 50:50 beam splitting condition. These results constitute an important step towards programmable routing and entanglement distribution in large-scale quantum networks and distributed quantum computers.