We determine for the first time the magnetic dipole moment of a short-lived nucleus with part-per-million (ppm) accuracy. To achieve this 2-orders-of-magnitude improvement over previous studies, we implement a number of innovations into ourβ-detected nuclear magnetic resonance (β-NMR) setup at ISOLDE at CERN. Using liquid samples as hosts, we obtain narrow, subkilohertz-linewidth, resonances, while a simultaneous in situH1NMR measurement allows us to calibrate and stabilize the magnetic field to ppm precision, thus eliminating the need for additionalβ-NMR reference measurements. Furthermore, we use ab initio calculations of NMR shielding constants to improve the accuracy of the reference magnetic moment, thus removing a large systematic error. We demonstrate the potential of this combined approach with the 1.1 s half-life radioactive nucleusNa26, which is relevant for biochemical studies. Our technique can be readily extended to other isotopic chains, providing accurate magnetic moments for many short-lived nuclei. Furthermore, we discuss how our approach can open the path toward a wide range of applications of the ultrasensitiveβ-NMR in physics, chemistry, and biology.