This work deals with observer-based output-feedback stabilization of linear time-varying (LTV) systems with time-varying measurement delay. The delay is modeled by a first-order hyperbolic partial differential equation (PDE), whose boundary condition evolves according to an ordinary differential equation (ODE), namely, the plant. The overall system is thus represented by an ODE-PDE cascade, which is also the structure of the proposed observer. The exponential stability of the estimation error is established for arbitrarily large time-varying delays. An observer-based controller is also introduced and the closed-loop stability is proved. The separation principle holds and the design of both the observer and controller gains can be done as if there was no delay. Some simulations illustrate the feasibility of this approach. (C) 2019 Elsevier Ltd. All rights reserved.