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 similar to 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 similar to 0.05 mu G are required to be uniform over scales of similar to 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.