Within the framework of the project “C³ – Carbon Concrete Composite” the composite material carbon reinforced concrete and its application for constructions has been researched. The “CUBE” building, which is a C³ technology demonstrations house, exemplifies the results of this project. It is assumed to be the world’s first building including concrete components made exclusively with non-metallic, mainly with carbon reinforcement. The house is erected at the campus of the Technische Universität Dresden, east Germany and will be completed in 2022. The main purpose of the building is to show, that the C³ construction method has progressed so far, that the entire process chain can be passed through, including the design, planning, tendering, construction and operation. Additionally, the building provides a testing facility site where innovative components of multiple future research projects can be implemented and tested under realistic conditions. This paper presents this very challenging and trend-setting building project. In detail, it deals with the architecture, the bearing structure, individual components and their special features. In particular, the high-performance building material carbon reinforced concrete and its application for the organically shaped, visually appealing and sustainable wall-roof structure of the CUBE-building is presented. Both the design of the wall-roof structure and its stepwise manufacturing are explained and illustrated with sketches and figures. Additionally, the resulting advantages over classical building materials are shown and discussed. All in all, the article about the design and manufacturing of the twisted roof-wall construction of the “CUBE” building shows a successful, practice-oriented application of carbon reinforced concrete and its diverse und huge usage potential for future, sustainable building constructions.