The ternary phase diagram for the silicone-based, nonionic surfactant L77-OH, water, and cyclohexane shows a narrow L2 region in the oil-rich corner. At 25-degrees-C the maximum water solubilization, as determined by the water-to-surfactant molar ratio, w(max), is 22.0. Since this is close to the estimated hydration number of the surfactant hydrophilic EO groups, reversed micellar aggregates, rather than water-in-oil microemulsion droplets, are believed to be present. Analysis of the SANS profiles from the dilute (phi < 0.05) L2 phase indicates that discrete lamellar aggregates are present at all w values. The SANS data are quantitatively fitted using a disk form factor P(Q) which determines the bilayer thickness (t) and allows an estimation of the lateral dimension through the disk radius (r). At low water content (w approximately 5.0) the value of t suggests that the EO chains are in an interdigitated (or coiled) configuration. It was found that t increases with w until at w(max) the EO chain is apparently fully extended. The radius also increases with w from 77 angstrom (w = 3.4) to larger dimensions (> 100 angstrom) which cannot accurately be measured using the available Q-range. Further support for a reversed bilayer structure in the L2 phase is obtained from studies of binary L77-OH/water mixtures. Polarizing microscopy shows the phase to be lamellar (L(alpha)). A similar layer thickness in the binary and ternary phases at the same w is found by SANS.