Motivated by the design of observers with good performance and robustness to measurement noise, the problem of estimating the state of a linear time-invariant system in finite time and robustly with respect to measurement noise is considered. Using a hybrid systems framework, a hybrid observer producing an estimate that converges to the plant state in finite time, even under unknown piecewise-constant noise, is presented. The stability and robustness properties of the observer are shown analytically and validated numerically. (C) 2015 Elsevier Ltd. All rights reserved.