With the use of Na-23 nuclear magnetic resonance spectroscopy and H-1 pulsed field gradient attenuation, we have determined the critical concentration at which the isotrope/nematic transition occurs within aqueous dispersions of synthetic Laponite clay used as a model of a charged anisotropic colloid. The dense macroscopic clay samples were prepared by oedometric compression, leading to a critical concentration for the isotrope/nematic transition significantly higher than that determined previously by using birefringence detection within capillary tubes. The increase of the nematic ordering detected by NMR as a function of the clay concentration is more progressive than the sharp ordering predicted by numerical simulations of the isotrope/nematic transition of rigid disks. This difference may result from some size and shape heterogeneities of the Laponite clay samples or from the competition between the relative orientation of the particles induced by the excluded volume and the electrostatic interactions.