We reconsider the non-Markovian time-continuous measurement of a Heisenberg observable x[over ] and show for the first time that it can be realized by an infinite set of entangled von Neumann detectors. The concept of continuous readout is introduced and used to rederive the non-Markovian stochastic Schrödinger equation. We can prove that, contrary to recent doubts, the resulting non-Markovian quantum trajectories are true single system trajectories and correspond to the continuous measurement of a retarded functional of x[over ].