We present multifrequency VLA observations of the intensity and polarized emission from G359.54+0.18, a system of nonthermal filaments near the Galactic Center. The intrinsic magnetic field lines run primarily along the filaments. The rotation measure (RM) varies between -4200 and -370 rad m^-2 on scales of several arcseconds, implying that the ionized, magnetized medium responsible for the Faraday rotation is less than 0.1 pc thick. In turn, this implies that the magnetic field in the Faraday screen is large, suggesting that it is located close to the Galactic center. Further evidence is provided by the anisotropy of the RM fluctuations. The structure of the eastern portion of G359.54+0.18 suggests that the magnetic field in the filaments is highly distorted as a result of an interaction with an adjacent Galactic center molecular cloud.