The production process of the invention comprises a first step of hydrodesulfurizing of catalytically-cracked gasoline so as to result in an olefin hydrogenation rate of no greater than 25 mol % in the catalytically-cracked gasoline, a total sulfur content of no greater than 20 ppm by weight based on the product oil weight, a sulfur content derived from thiophenes and benzothiophenes of no greater than 5 ppm by weight and a sulfur content derived from thiacyclopentanes of no greater than 0.1 ppm by weight, and a second step of further hydrodesulfurizing of the product oil obtained by the first step so as to result in a total of no greater than 30 mol % for the olefin hydrogenation rate in the first step and the olefin hydrogenation rate in the second step, a total sulfur content of no greater than 10 ppm by weight based on the product oil weight, and a sulfur content derived from thiols of no greater than 5 ppm by weight.