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The present work proposes a new approach to the nonlinear observer design problem in the presence of delayed output measurements. The proposed nonlinear observer possesses a state-dependent gain which is computed from the solution of a system of first-order singular partial differential equations, and consists of a chain of state observation algorithms that reconstruct the unmeasurable state vector at different delayed time-instants within the time-delay window introduced by the output measurements. Therefore, the proposed nonlinear observer exhibits a chain-like structure that explicitly takes into account the magnitude of the output delay. Convergence of the estimation error to zero is established for an arbitrary delay in the output measurements.