This paper provides a method for designing robust state observers for a class of time-delay systems where unknown exogenous disturbances and time delays appear in both the states and the output measurements. First, by introducing a new integral output vector, we transform the considered time-delay system into an augmented system where the unknown disturbance vector is treated as the unknown input vector and the output measurement vector is delay-free and disturbance-free. Second, we propose a new coordinate state transformation method to transform the augmented system into an observable canonical form where time delays appear only in the input and output vectors. Third, we design a robust state observer to robustly estimate the state vector such that the observer error vector converges within a ball with the smallest possible radius. Fourth, we consider the problem of reconstructing the original state vector of the system. Finally, three examples are provided to support the proposed method.