Dispersions of colloidal particles in cholesteric liquid crystals form an unusual solid by stabilizing a network of linear defects under tension in the ideal layered structure of the cholesteric. The large length scales of the cholesteric liquid crystals allowed direct observation of the network structure, and its properties were correlated with rheological measurements of elasticity. This system serves as a model for a class of solids formed when particles are mixed with layered materials such as thermotropic and lyotropic smectic liquid crystals and block copolymers.