A single nuclear spin holds the promise of being a long-lived quantum bit or quantum memory, with the high fidelities required for fault-tolerant quantum computing. We show here that such promise could be fulfilled by a single phosphorus (31P) nuclear spin in a silicon nanostructure. By integrating single-shot readout of the electron spin with on-chip electron spin resonance, we demonstrate the quantum non-demolition, electrical single-shot readout of the nuclear spin, with readout fidelity better than 99.8% - the highest for any solid-state qubit. The single nuclear spin is then operated as a qubit by applying coherent radiofrequency (RF) pulses. For an ionized 31P donor we find a nuclear spin coherence time of 60 ms and a 1-qubit gate control fidelity exceeding 98%. These results demonstrate that the dominant technology of modern electronics can be adapted to host a complete electrical measurement and control platform for nuclear spin-based quantum information processing.