The director-field configuration of a nematic liquid crystal confined to cylindrical cavities of polycarbonate Nuclepore membranes ranging from 0.3 to 0.05-mu-m in radius is determined using deuterium nuclear magnetic resonance (H-2 NMR). Spectral patterns from cavities of radius 0.3-mu-m reveal the escaped-radial configuration with singular point defects, but as the cylinder size is decreased, the elastic energy imposed by the curvature of the confining walls competes with the anchoring energy to tilt the directors away from their preferred perpendicular anchoring direction, preventing the expected transition to the planar-radial configuration. A surface fitting parameter is directly determined by simulating H-2-NMR line shapes, and by studying a series of samples with different radii, the molecular-anchoring strength W0 and surface elastic constant K24 are extracted.