In this paper, we study the quadratic regulator problem for a process governed by a Volterra integrodifferential equation in R-n. Our main goal is the proof that it is possible to associate a Riccati differential equation to this quadratic control problem, from which the feedback form of the optimal control can be computed. This is in contrast with previous papers on the subject, which confine themselves to study the Fredholm integral equation which is solved by the optimal control.