We consider the pursuit problem in 2-person differential game, one player is a pursuer and another one is an evader. The problem is given by a system of differential–difference equations with time lag. The players choose their controls in the form of measurable functions with values from certain compacts. The goal of the pursuer is to catch the evader in the shortest possible time. The goal of the evader is to avoid the meeting of the players’ trajectories on a whole semiinfinite interval of time or if it is impossible to maximally postpone the moment of meeting. For such a conflict-controlled process we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain moment of time for any counteractions of the evader.