A detailed line shape analysis of Na-23 nuclear magnetic resonance spectra within dense suspensions (12% vol/vol) of Laponite, clay exhibits a macroscopic ordering of these charged anisotropic colloids within a nematic phase. The angular variation of the order parameter limits to 20% of the maximum amount of disorder in these dense suspensions. By contrast, dilute Laponite suspensions (1-4% vol/vol) remain isotropic while the variation of Na-23 relaxation rates over a broad range of frequencies indicates a local ordering of the clay platelets within microdomains of the same spatial extent than the particle diameter (300 Angstrom).