All-solid-state Li-ion batteries with metallic Li anodes and solid electrolytes could offer superior energy density and safety over conventional Li-ion batteries. However, compared with organic liquid electrolytes, the low conductivity of solid electrolytes and large electrolyte/electrode interfacial resistance impede their practical application. Garnet-type Li-ion conducting oxides are among the most promising electrolytes for all-solid-state Li-ion batteries. In this work, the large-radius Rb is doped at the La site of cubic Li6.10Ga0.30La3Zr2O12 to enhance the Li-ion conductivity for the first time. The Li6.20Ga0.30La2.95Rb0.05Zr2O12 electrolyte exhibits a Li-ion conductivity of 1.62 mS cm-1 at room temperature, which is the highest conductivity reported until now. All-solid-state Li-ion batteries are constructed from the electrolyte, metallic Li anode, and LiFePO4 active cathode. The addition of Li(CF3SO2)2N electrolytic salt in the cathode effectively reduces the interfacial resistance, allowing for a high initial discharge capacity of 152 mAh g-1 and good cycling stability with 110 mAh g-1 retained after 20 cycles at a charge/discharge rate of 0.05 C at 60 °C.