Mahler  and Fejes Tóth  proved that every centrally symmetric convex plane bodyK admits a packing in the plane by congruent copies ofK with density at least √3/2. In this paper we extend this result to all, not necessarily symmetric, convex plane bodies. The methods of Mahler and Fejes Tóth are constructive and produce lattice packings consisting of translates ofK. Our method is constructive as well, and it produces double-lattice packings consisting of translates ofK and translates of−K. The lower bound of √3/2 for packing densities produced here is an improvement of the bounds obtained previously in  and .