In many biological and environmental studies, measured data is subject to a limit of detection. The limit of detection is generally defined as the lowest concentration of analyte that can be differentiated from a blank sample with some certainty. Data falling below the limit of detection is left-censored, falling below a level that is easily quantified by a measuring device. A great deal of interest lies in estimating the limit of detection for a particular measurement device. In this paper we propose a change-point model to estimate the limit of detection using data from an experiment with known analyte concentrations. Estimation of the limit of detection proceeds by a two-stage maximum likelihood method. Extensions are considered that allow for censored measurements and data from multiple experiments. A simulation study is conducted demonstrating that in some settings the change-point model provides less biased estimates of the limit of detection than conventional methods. The proposed method is then applied to data from an HIV pilot study.