Aqueous dispersions of nanometric clay platelets (Laponite) have been dewatered through different techniques: centrifugation, mechanical compression, and osmotic stress (dialysis against a polymer solution). The positional and orientational correlations of the particles have been determined through small-angle neutron scattering. Uniaxial compression experiments produce concentrated dispersions (volume fraction > 0.03) in which the platelets have strong orientational and positional correlations. The orientational correlations cause the platelets to align with their normal along a common axis, which is the axis of compression. The positional correlations cause the platelets to be regularly spaced along this direction, with a spacing that matches the average volume per particle in the dispersion. The swelling law (volume fraction versus separation distance) is one-dimensional, as in a layered system. Changes in the applied osmotic pressure cause the water content of the dispersion to either rise or decrease, with time scales that are controlled by interparticle friction forces and by hydrodynamic drag. At long times, the dispersions approach osmotic equilibrium, which can be defined as the common limit of swelling and deswelling processes. The variation of the equilibrium water content with the applied osmotic pressure has been determined over 1 decade in volume fractions (0.03 < phi < 0.3) and 3 decades in pressures. This equation of state matches the predictions made from the knowledge of the forces and thermal agitation for all components in the dispersion (particles, ions, and water).