Multifrequency VLA observations of the thin filamentary structure, G359.54+0.18 reveal the magnetic field geometry and detailed rotation measure and spatial distributions of this system of nonthermal filaments. The intrinsic magnetic field lines run primarily along the filaments. The rotation measure (RM) varies between -4200 and -370 rad m^-2 along the extent of the filaments, 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, which are stronger in the direction running along the filament. 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.