Abstract: We consider a finite-horizon two-person zero-sum differential game in which the system dynamics is described by a linear differential equation with a Caputo fractional derivative and the goals of the players’ control are to minimize and maximize a quadratic terminal-integral cost function, respectively. We present conditions for the existence of a game value and obtain formulas for players’ optimal feedback control strategies with memory of motion history. The results are based on the construction of a solution to an appropriate Hamilton–Jacobi equation with fractional coinvariant derivatives under a natural right-end boundary condition. © Pleiades Publishing, Ltd. 2023. ISSN 1064-5624, Doklady Mathematics, 2023, Vol. 108, Suppl. 1, pp. S122–S127. Pleiades Publishing, Ltd., 2023. Russian Text The Author(s), 2023, published in Matematicheskaya Teoriya Igr i Ee Prilozheniya, 2023, Vol. 15, No. 2, pp. 18–32.