We consider the problem of differentially private selection. Given a finite set of candidate items, and a quality score for each item, our goal is to design a differentially private mechanism that returns an item with a score that is as high as possible. The most commonly used mechanism for this task is the exponential mechanism. In this work, we propose a new mechanism for this task based on a careful analysis of the privacy constraints. The expected score of our mechanism is always at least as large as the exponential mechanism, and can offer improvements up to a factor of two. Our mechanism is simple to implement and runs in linear time.