The identification of fluorescently stained cell nuclei is the basis of cell detection, segmentation, and feature extraction in high content microscopy experiments. The nuclear morphology of single cells is also one of the essential indicators of phenotypic variation. However, the cells used in experiments can lose their contact inhibition, and can therefore pile up on top of each other, making the detection of single cells extremely challenging using current segmentation methods. The model we present here can detect cell nuclei and their morphology even in high-confluency cell cultures with many overlapping cell nuclei. We combine the "gas of near circles" active contour model, which favors circular shapes but allows slight variations around them, with a new data model. This captures a common property of many microscopic imaging techniques: the intensities from superposed nuclei are additive, so that two overlapping nuclei, for example, have a total intensity that is approximately double the intensity of a single nucleus. We demonstrate the power of our method on microscopic images of cells, comparing the results with those obtained from a widely used approach, and with manual image segmentations by experts.