In accordance with the principles of sustainable development, environmentally friendly, low-emission, and energy-intensive materials and technologies, as well as waste management, should be used. Concrete production is responsible for significant energy consumption and CO2 production; therefore, it is necessary to look for new solutions in which components are replaced by other materials, preferably recycled. A positive way is to use glass waste. In order to determine the effect of a significant glass cullet content on the properties of concrete, glass powder was used as a filler and 100% glass aggregate. The cement–glass composite has low tensile strength and brittle failure. In order to improve tensile strength, the effects of adding polypropylene fibres on the mechanical properties of the composite were investigated. With the addition of 300, 600, 900, 1200, and 1500 g/m3 of fibres, which corresponds to 0.0625%, 0.1250%, 0.1875%, 0.2500%, and 0.3125% of cement mass, respectively, flexural strength increased compared with the base sample by 4.1%, 8.2%, 14.3%, 20.4%, and 26.5%, respectively, while the increase in splitting strength was 35%, 45%, 115%, 135%, and 185%, respectively. Moreover, with the addition of fibres, a decrease in slump by 25.9%, 39.7%, 48.3%, 56.9%, and 65.5%, respectively, compared with the reference specimen was determined.