A lithium-sulfur battery comprises a cathode electrode comprising from 80% to 100% lithium polysulfide based on the total weight of sulfur adsorbed at the cathode when the lithium sulfur battery is fully charged, and a high specific surface area electrically conductive material. An anode electrode comprises lithium. A porous and electrically insulating membrane is provided between the cathode and the anode electrodes. An electrolyte is adsorbed into and between cathode electrode, the anode electrode, and the membrane. A cathode current collector is electrically connected to the cathode and an anode current collector is electrically connected to the anode. A porous and electrically conductive interlayer can be provided between the membrane and at least one selected from the group consisting of the cathode and the anode. A method of making a battery is also disclosed.