Stoichoimetric graphene fluoride monolayers are obtained in a single step by the liquid-phase exfoliation of graphite fluoride with sulfolane. Comparative quantum-mechanical calculations reveal that graphene fluoride is the most thermodynamically stable of five studied hypothetical graphene derivatives; graphane, graphene fluoride, bromide, chloride, and iodide. The graphene fluoride is transformed into graphene via graphene iodide, a spontaneously decomposing intermediate. The calculated bandgaps of graphene halides vary from zero for graphene bromide to 3.1 eV for graphene fluoride. It is possible to design the electronic properties of such two-dimensional crystals.