Observer design for linear systems with aperiodic sampled-data measurements is addressed. To solve this problem, a novel hybrid observer is designed. The main peculiarity of the proposed observer consists of the use two output injection terms, one acting at the sampling instants and one providing an intersample injection. The error dynamics are augmented with a timer variable triggering the arrival of a new measurement and analyzed via hybrid system tools. Using Lyapunov theory, sufficient conditions for the convergence of the observer are provided. Relying on those conditions, an optimal LMI-based design is proposed for the observer gains. The effectiveness of the approach is illustrated in an example.