A solid Li+ electrolyte, which has a high Li+ conductivity at 25 degrees C and a high density, is required in rechargeable Li-ion batteries including all-solid-state Li-metal batteries and Li-redox flow batteries. Here we prepare a fast cubic perovskite Li+ conductor, Li3/8Sr7/16Ta3/4Hf1/4O3 (LSTH), by reducing the grain boundary resistance with spark plasma sintering. The influence of the sintering temperature and time on the relative density and the total Li+ conductivity of Li3/8Sr7/16Ta3/4Hf1/4O3 are investigated; the LSTH pellets sintered by SPS at 1250 degrees C for 10 min has a much higher density (96.5%), a higher Li+ conductivity (5.2 x 10(-4) S cm(-1)) at 25 degrees C and a lower activation energy (0.33 eV) than the Li3/8Sr7/16Ta3/4Hf1/4O3 pellet fired with conventional sintering. The symmetric Li/Li cell with a LSTH membrane, which is coated by a thin Li+-conducting solid polymer to avoid the chemical reduction of Li3/8Sr7/16Ta3/4Hf1/4O3 by a metallic lithium anode and to homogenize the current density at Li/electrolyte interface, shows a stable cycling at 0.3 mA cm(-2); the all-solid-state Li/LiFePO4 battery and the Li-S cell with Li3/8Sr7/16Ta3/4Hf1/4O3 membranes show a high coulombic efficiency and a good cycling life.