The microscopic structure of a microemulsion of triolein oil, aqua (water-ethanol, 80/20 wt %), and polyoxyethylene (40) sorbitol hexaoleats has been studied with small-angle neutron scattering. Contrast variation (in the scattering of neutrons) is accomplished by the deuteration of the aqua and oil components of the system. Samples for which only the aqueous phase is deuterated reveal the geometry of the aqueous phase, and those in which only the oil is deuterated reveal the geometry of the oil phase. Finally, when both oil and aqua are contrast matched by deuteration, the surfactant geometry is measured. The aqueous-surfactant surface and the oil-surfactant surface are found to be of significantly different geometries. These two surfaces are described in detail for varying aqua concentration by the very successful leveled-wave model. The behavior of the leveled-wave model parameters produced by the aqua geometry, as a function of aqua concentration and temperature, correlates with a thermodynamic instability in the miscibility. Further, the spectrum of neutrons produced by the surfactant itself is different from the aqua and oil spectra. Although a detailed geometric description of the surfactant is not available for this component (no geometric model being applicable), the data are consistent with there being different geometries for the two bounding surfaces. A plausible correlation between the geometric behavior of the material and a miscibility gap is also detected.