NOVELTY - Manufacture of garnet-type compound involves obtaining a precipitate by mixing raw material (a) containing zirconium carbonate complex, and raw material (b) containing compound containing element (M1) chosen from lanthanum, scandium, yttrium, and cerium as a constituent element, and obtaining a baked product by baking mixture at temperature of less than 1000 degrees C after preparing mixture by mixing the precipitate with raw material (c) containing compound containing lithium as constituent element. The zirconium carbonate complex contains nitrogen-containing counter ion (I).    USE - Manufacture of garnet-type compound used for forming electrode layer of all-solid-state lithium secondary battery (claimed). Can also be used for fuel cell and gas sensor.    ADVANTAGE - The method enables manufacture of garnet-type compound having high lithium-ion conductivity, with high production efficiency and suppressed outflow of water-soluble lithium component, without using an organic solvent, providing lithium secondary battery with high rate characteristics and energy density, and suppressed internal resistance.    DETAILED DESCRIPTION - Manufacture of garnet-type compound involves obtaining a precipitate by mixing a raw material (a) containing zirconium carbonate complex, and raw material (b) containing compound containing element (M1) chosen from lanthanum, scandium, yttrium, and cerium as a constituent element, obtaining a baked product by baking a mixture at a temperature of less than 1000 degrees C after preparing the mixture by mixing the precipitate with raw material (c) containing compound containing lithium as constituent element, and obtaining garnet-type compound containing zirconium, lithium, and element (M1). The raw material (a) is a solution having a pH of 7-9.5. The molar ratio of zirconium component with respect to carbonate component contained in the raw material (a) is 1.5-15. The zirconium carbonate complex contains nitrogen-containing counter ion of formula: N(R4)+ (I).    R=substituent chosen from H, CH3, or CH2CH2OH.    An INDEPENDENT CLAIM is included for all-solid-state lithium secondary battery, which consists of a positive electrode layer containing a positive electrode active material, a negative electrode layer containing a negative electrode active material, and a solid electrolyte layer provided between the positive electrode layer and the negative electrode layer. The positive electrode layer, negative electrode layer, or solid electrolyte layer contains the garnet-type compound.