A rechargeable battery cell has an organic-liquid electrolyte contacting a dendrite free alkali-metal anode. The alkali-metal anode may be a liquid at the operating temperature that is immobilized by absorption into a porous membrane. The alkali-metal anode may be a solid that wets a porous-membrane separator, where the contact between the solid alkali-metal anode and the liquid electrolyte is at micropores or nanopores in the porous-membrane separator. The use of a dendrite-free solid lithium cell was demonstrated in a symmetric cell with a porous cellulose-based separator membrane. A K+-ion rechargeable cell was demonstrated with a liquid K—Na alloy anode immobilized in a porous carbon membrane using an organic-liquid electrolyte with a Celgard® or glass-fiber separator.