We address the problem of constructing asymptotic observers for linear time-invariant systems with measurements subject to outliers. We propose a linear observer with a saturated output injection, which allows one to roughly mitigate the effect of abnormal and isolated measurement errors on the error dynamics. In addition to proposing the observer architecture, we provide convex conditions on the observer parameters that ensure uniform global exponential stability of the origin for the error dynamics. Such conditions can be employed for optimal parameter selection by using linear matrix inequalities. The effectiveness of the approach is shown by means of some numerical examples.