The ability to communicate quantum information over long distances is of central importance in quantum science and engineering1. Although some applications of quantum communication such as secure quantum key distribution2,3 are already being successfully deployed4-7, their range is currently limited by photon losses and cannot be extended using straightforward measure-and-repeat strategies without compromising unconditional security8. Alternatively, quantum repeaters9, which utilize intermediate quantum memory nodes and error correction techniques, can extend the range of quantum channels. However, their implementation remains an outstanding challenge10-16, requiring a combination of efficient and high-fidelity quantum memories, gate operations, and measurements. Here we use a single solid-state spin memory integrated in a nanophotonic diamond resonator17-19 to implement asynchronous photonic Bell-state measurements, which are a key component of quantum repeaters. In a proof-of-principle experiment, we demonstrate high-fidelity operation that effectively enables quantum communication at a rate that surpasses the ideal loss-equivalent direct-transmission method while operating at megahertz clock speeds. These results represent a crucial step towards practical quantum repeaters and large-scale quantum networks20,21.