When produced under reducing conditions, beta-Ga2O3 is transformed into an n-type semiconductor with delocalized conduction electrons that exhibit a very strong electron spin resonance (ESR) and a strong hyperfine coupling to the gallium nuclei of the host lattice. We apply the Overhauser-shift technique to investigate single crystals of this compound. With extension to the high magnetic field of a W-band spectrometer, we were able to resolve all spectral lines that were recorded and to assign them to their corresponding electronic and nuclear states. This separate analysis was the basis to access additional sample characteristics: the hyperfine coupling that is actually averaged out in the ESR signal, as well as the nuclear relaxation rates could be analyzed. Systematic measurements by varying the microwave power revealed the Overhauser shift in thermal equilibrium. The signal could be tracked to very small microwave saturation parameters, at which the deviation from the usual linear relation between power and shift becomes evident and the shift clearly approaches a constant value. This value in equilibrium was determined directly from a fit to a sequence of measurements, whereas standard X-band experiments only provided indirect conclusions. The probability densities of the electrons at the nuclei in the two nonequivalent crystallographic positions-the lattice sites with octahedral and tetrahedral coordination-could also be determined directly. The enhanced resolution revealed an otherwise hidden substructure in the nuclear resonance signals. On the basis of a microscopic model, this structure could be used to probe the environment of the oxygen vacancy more precisely and to determine the extension of the electronic wave function of the donor electrons.