NMR studies of xenon gas dissolved in the liquid crystal ZLI 1132 confined to submicron cylindrical cavities are reported. Spectra taken as a function of temperature yield a clear indication of the nematic to isotropic phase transition of the confined liquid crystals. In the nematic phase at 21 degrees C, the resonance line of dissolved Xe-129 exhibits a chemical shift anisotropy of 15 ppm due to a random distribution of director axes in the plane perpendicular to the long axis of the cylinder. The anisotropy and temperature dependence of the confined system are compared to control experiments that use the bulk liquid crystal. The quadrupolar splitting observed in the Xe-131 NMR spectrum of the confined liquid crystalline solution of xenon gas is slightly greater than that found in the bulk. Two-dimensional exchange NMR demonstrates that the xenon atoms probe different average liquid crystal directors within a single cavity on a 20 ms time scale and that interpore exchange occurs on a time scale of 400 ms. The exchange data indicate that changes in the orientation of the director within individual cavities occur on a length scale of about 2 mu m.