Quantum systems that consist of solid-state electronic spins can be sensitive detectors of nuclear magnetic resonance (NMR) signals, particularly from very small samples. For example, nitrogen-vacancy centres in diamond have been used to record NMR signals from nanometre-scale samples(1-3), with sensitivity sufficient to detect the magnetic field produced by a single protein(4). However, the best reported spectral resolution for NMR of molecules using nitrogen-vacancy centres is about 100 hertz(5). This is insufficient to resolve the key spectral identifiers of molecular structure that are critical to NMR applications in chemistry, structural biology and materials research, such as scalar couplings (which require a resolution of less than ten hertz(6)) and small chemical shifts (which require a resolution of around one part per million of the nuclear Larmor frequency). Conventional, inductively detected NMR can provide the necessary high spectral resolution, but its limited sensitivity typically requires millimetre-scale samples, precluding applications that involve smaller samples, such as picolitre-volume chemical analysis or correlated optical and NMR microscopy. Here we demonstrate a measurement technique that uses a solid-state spin sensor (a magnetometer) consisting of an ensemble of nitrogen-vacancy centres in combination with a narrowband synchronized readout protocol(7-9) to obtain NMR spectral resolution of about one hertz. We use this technique to observe NMR scalar couplings in a micrometre-scale sample volume of approximately ten picolitres. We also use the ensemble of nitrogen-vacancy centres to apply NMR to thermally polarized nuclear spins and resolve chemicalshift spectra from small molecules. Our technique enables analytical NMR spectroscopy at the scale of single cells.