Multifrequency polarimetry with the Very Long Baseline Array telescope has revealed absolute Faraday rotation measures (RMs) in excess of 1000 rad m-2 in the central regions of the quasars 3C 273, 3C 279, and 3C 380. Beyond a projected distance of ~20 h-1 pc, however, the jets of all three sources are found to have RM < 100 rad m-2. Such sharp RM gradients cannot be produced by cluster or galactic-scale magnetic fields but rather must be the result of magnetic fields organized on scales of 1-100 pc. If associated with the region that produces the narrow optical emission lines, then magnetic fields of strength ~0.05 μG are required to be uniform over scales of ~10 pc. The direct detection of high RMs in these quasar cores can explain the low fractional-core polarizations usually observed in quasars at centimeter wavelengths as the result of irregularities in the Faraday screen on scales smaller than the telescope beam.