We are developing automated analysis of corneal-endothelial-cell-layer, specular microscopic images so as to determine quantitative biomarkers indicative of corneal health following corneal transplantation. Especially on these images of varying quality, commercial automated image analysis systems can give inaccurate results, and manual methods are very labor intensive. We have developed a method to automatically segment endothelial cells with a process that included image flattening, U-Net deep learning, and postprocessing to create individual cell segmentations. We used 130 corneal endothelial cell images following one type of corneal transplantation (Descemet stripping automated endothelial keratoplasty) with expert-reader annotated cell borders. We obtained very good pixelwise segmentation performance (e.g., Dice coefficient  =  0.87  ±  0.17, Jaccard index  =  0.80  ±  0.18, across 10 folds). The automated method segmented cells left unmarked by analysts and sometimes segmented cells differently than analysts (e.g., one cell was split or two cells were merged). A clinically informative visual analysis of the held-out test set showed that 92% of cells within manually labeled regions were acceptably segmented and that, as compared to manual segmentation, automation added 21% more correctly segmented cells. We speculate that automation could reduce 15 to 30 min of manual segmentation to 3 to 5 min of manual review and editing.