The level-mixing resonance technique applied to beta-decaying nuclei (beta-LMR) is a well-established tool for measuring the ground state quadrupole moments of nuclei away from stability. These experiments yield the quadrupole to magnetic moment ratio, provided the electric field gradient (EFG) of the implanted nuclei in the crystal is known. By combining beta-LMR with beta nuclear magnetic resonance (beta-NMR), both the quadrupole moment Q and the magnetic moment mu can be simultaneously extracted in a single experiment. A major advantage of this technique is that the initial nuclei need only to be aligned, and not necessarily polarized. Alignment is generally easier to produce than polarization, and occurs at the highest yield point in the nuclear reaction mechanism. This is an especially important consideration in the study of dipole and quadrupole moments of nuclei in the region of the drip lines. We report here the successful application of this combined technique to the known case of B-12(Mg). The main features of the LMR-NMR technique are described. [S0556-2813(99)01204-2].