The magnetic field is the key physical quantity responsible for the formation, stability, and evolution of solar prominences (ribbons of cool dense gas embedded in the hot tenuous corona). Therefore, it is important to obtain good empirical knowledge of the three-dimensional structure of prominence magnetic fields. Here we show how the magnetic field vector can be inferred via the physical interpretation of spectropolarimetric observations in the He I λ10830 multiplet. To this end, we have developed an inversion code based on the quantum theory of the Hanle and Zeeman effects and on a few modeling assumptions. We show an application to full Stokes vector observations of a polar crown prominence that, in the slit-jaw Hα image, showed nearly vertical plasma structures. Our results provide evidence for magnetic fields on the order of 30 G inclined by about 25° with respect to the local solar vertical direction. Of additional interest is that the inferred nearly vertical magnetic field vector appears to be slightly rotating around a fixed direction in space as one proceeds along the direction of the spectrograph's slit. While these results provide new light on the three-dimensional geometry of the magnetic fields that confine the plasma of polar crown prominences, they also urge us to develop improved solar prominence models and to pursue new diagnostic investigations.