The problem of state estimation for nonlinear systems with unknown state or measurement delays is still an open problem. In this paper, we consider the case of measurement delay and propose an approach that combines a delay identifier with a suitable high-gain observer in order to achieve simultaneous estimation of state and delay. We provide sufficient conditions that guarantee the exponential convergence to zero of the errors, globally with respect to the system variables and locally with respect to the delay estimation. We validate the method through an example concerning population models.