We report a stoichiometric derivative of graphene with a fluorine atom attached to each carbon. Raman, optical, structural, micromechanical and transport studies show that the material is qualitatively different from the known graphene-based nonstoichiometric derivatives. Fluorographene is a high-quality insulator (resistivity >10^12 Ohm per square) with an optical gap of 3 eV. It inherits the mechanical strength of graphene, exhibiting Youngs modulus of 100 N/m and sustaining strains of 15%. Fluorographene is inert and stable up to 400C even in air, similar to Teflon.