An attractive diagnostic tool for investigating the magnetism of the solar chromosphere is the observation and theoretical modeling of the Hanle and Zeeman effects in spicules, as shown in this Letter for the first time. Here we report on spectropolarimetric observations of solar chromospheric spicules in the He I λ10830 multiplet and on their theoretical modeling accounting for radiative transfer effects. We find that the magnetic field in the observed (quiet-Sun) spicular material at a height of about 2000 km above the visible solar surface has a strength of the order of 10 G and is inclined by approximately 35deg with respect to the local vertical direction. Our empirical finding based on full Stokes vector spectropolarimetry should be taken into account in future magnetohydrodynamical simulations of spicules.