A process for producing sulfuric acid and cement clinker and for obtaining rare earths using calcium sulfate which is formed and separated off as a solid by-product in production of phosphoric acid in the reaction of raw phosphate with sulfuric acid to form phosphoric acid, wherein the process comprises the following steps: a) the calcium sulfate separated from the phosphoric acid is treated with an acid in order to obtain a suspension comprising purified calcium sulfate, b) the purified calcium sulfate is separated in solid form from the liquid phase of the suspension obtained, b1) the purified calcium sulfate obtained in step b) is treated with water or with a salt- and/or chelate ligand-containing aqueous solution, in order to leach the rare earths out of the calcium sulfate, b2) the further-purified calcium sulfate is separated in solid form from the liquid phase of the suspension obtained in step b1), with one or more rare earths present in the liquid phase, c) the purified calcium sulfate separated off in step b1) is mixed with admixtures and reducing agents in order to obtain a raw meal mixture for cement clinker production, d) the raw meal mixture is burnt in order to obtain the cement clinker, with formation of sulfur dioxide as offgas, and e) the sulfur dioxide formed is fed as raw material to sulfuric acid production in order to produce the sulfuric acid, it being possible to use the sulfuric acid produced as starting material in phosphoric acid production and/or send it to another use, characterized in that the weight ratio of solids to liquid (S/L ratio) for the leaching of the rare earths out of the phosphogypsum in step b1) is 1:1 to 1:10, where the purified calcium sulfate separated off which is obtained in step b) contains at least 50% by weight of anhydrite, based on dry calcium sulfate, where the dwell time for the leaching of the rare earths out of the phosphogypsum in step b1) is 5 h to 68 h. a) the calcium sulfate separated from the phosphoric acid is treated with an acid in order to obtain a suspension comprising purified calcium sulfate, b) the purified calcium sulfate is separated in solid form from the liquid phase of the suspension obtained, b1) the purified calcium sulfate obtained in step b) is treated with water or with a salt- and/or chelate ligand-containing aqueous solution, in order to leach the rare earths out of the calcium sulfate, b2) the further-purified calcium sulfate is separated in solid form from the liquid phase of the suspension obtained in step b1), with one or more rare earths present in the liquid phase, b1) the purified calcium sulfate obtained in step b) is treated with water or with a salt- and/or chelate ligand-containing aqueous solution, in order to leach the rare earths out of the calcium sulfate, b2) the further-purified calcium sulfate is separated in solid form from the liquid phase of the suspension obtained in step b1), with one or more rare earths present in the liquid phase, c) the purified calcium sulfate separated off in step b1) is mixed with admixtures and reducing agents in order to obtain a raw meal mixture for cement clinker production, d) the raw meal mixture is burnt in order to obtain the cement clinker, with formation of sulfur dioxide as offgas, and e) the sulfur dioxide formed is fed as raw material to sulfuric acid production in order to produce the sulfuric acid, it being possible to use the sulfuric acid produced as starting material in phosphoric acid production and/or send it to another use,