The domain growth processes occurring during the gravity-dominated regime of the phase separation of a cyclohexane/aniline critical mixture, following a temperature quench in the unstable region below the temperature-composition coexistence curve, were investigated with high-resolution H-1 NMR. Diffusive exchange of cyclohexane molecules between a small volume fraction of immiscible droplets and the surrounding fluid with a different cyclohexane concentration leads to a domain-size dependent H-1 NMR cyclohexane chemical shift in the continuous phase. In this case, the equations of motion for the transverse magnetization density reduce to a first-order exchange between two immiscible phases with rate coefficients depending on the droplet size and the transport process across the liquid-liquid interface. Application of this model to the data yields a power law for the growing droplets, R(t) = At-phi, where R is a typical domain size, with the growth exponent phi=0.30+/-0.01 and the amplitude A congruent to 0.9X10(-6) ms(-phi). These values are in close ms agreement with previous visualization studies on the same binary mixture [F. Cau and S. Lacelle, Phys. Rev. E 47, 1429 (1993)]. The results also confirm the presence of a new domain growth regime involving Brownian coagulation and sedimentation in the very late stage of the phase separation processes of binary liquid mixtures. A strong linear composition dependence of the H-1 NMR cyclohexane chemical shift in one-phase cyclohexane/aniline mixtures was interpreted on the basis of solvent shift effects induced by intermolecular interactions and susceptibility effects. In each phase of the two-phase equilibrium mixture, inhomogeneous line broadening, arising from the magnetic susceptibility discontinuity at the interface between phases, is also investigated.