The present paper reports on dispersions of intemally liquid crystalline particles, formed from monoglyceride and oil mixtures, stabilized with discrete disklike particles of Laponite clay. Small-angle X-ray scattering (SAXS) was used to probe the presence of dispersed particles as well as their internal liquid crystalline structure. The data were compared to scattering results of reference systems, namely, from the bulk as well as from well-defined particles formed with a polymer as the emulsifier. The submicrometer sizes of the various particles could be derived using dynamic light scattering (DLS). The possible mechanisms involved in the stabilization of each of the different phases by the Laponite platelets, including the role of the residual salt, are discussed. Time-resolved experiments were performed over 60 days in order to follow the evolution of both the internal structure and size of the particles. In particular, we discuss the peculiar behavior of the sample without added oil, where the cubosomes transform into hexosomes over time. The effect of the high pH induced by the Laponite platelets in water, which could result in a hydrolysis of the monoglycerides, was shown to be responsible for the observed cubosome-to-hexasome transition, as well as for the decrease in the lattice parameters.