Among Bricard’s overconstrained 6R linkages, the third type has two collapsed configurations, where all joint axes are coplanar. The paper presents a one-degree-of-freedom network of such linkages. Using the two coplanar states of the constituent Bricard units, the network is able to cover a large surface with a specific outline when deployed, and can be folded compactly into a stack of much smaller planar shapes. Five geometric parameters describing each type III Bricard mechanism are introduced. Their influence on the outline of one collapsed configuration is discussed and inverse calculation to obtain the parameter values yielding a desired planar shape is performed. The network is built by linking the units, either using scissor linkage elements, if the thickness of the panels can be ignored, or with hinged parallelograms, for a thicker material. Two case studies, in which the Bricard network deploys as a rectangle and a regular hexagon, respectively, are presented, validating the analysis and design methods.