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 control of the players are, respectively, to minimize and maximize a quadratic terminal-integral cost function. 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 basis of the results is the construction of a solution of the appropriate Hamilton"- Jacobi equation with so-called fractional coinvariant derivatives under a natural right-end boundary condition.