Dynamic light scattering (DLS) measurements of liquid-liquid demixing from dilute polystyrene (PS)/cyclohexane and PS/methylcyclohexane solutions are described. Phase transitions were induced by temperature quenches into the metastable region and beyond using concentrations low enough to ensure liquid-liquid demixing by a nucleation growth (NG) mechanism. In a companion paper the NG results are compared with demixing studies at higher concentration, where the mechanism is spinodal decomposition (SD), and with DLS measurements in a good solvent. The present measurements employed polystyrenes of low or medium molecular weight, M-w,M-1 = 30 000, rho(1) = M-w/M-n = 1.05, M-w,M-2 = 90 000, rho(2) = 1.04 and M-w,M-3 = 400 000, rho(3) = 1.05 In the NG region DLS measurements yield well-behaved correlation lengths for quenches well into the metastable region, but that pattern ends in the immediate vicinity of the cloudpoint where it splits to a bimodal distribution. The larger (slow) component is associated with the nucleation and growth of droplets of the precipitating phase. The rate of growth of the slow component was measured for the M-w 400 000 sample but growth was too rapid to follow at the lower M-w＇s. The splitting process is sensitive to both quench depth and quench rate. (At higher concentrations where liquid-liquid demixing is via an SD mechanism no splitting is observed. Instead, one finds well-behaved monomodal DLS correlograms which show exponentially increasing intensity and correlation length as the cloudpoint is approached.).