Monitoring the density of corneal endothelial cells (CEC) is essential in the management of corneal diseases. Its manual calculation is time consuming and prone to errors. U-Net, a neural network for biomedical image segmentation, has shown promising results in the automated segmentation of images of healthy corneas and good quality. The purpose of this study was to assess its performance in "real-world" CEC images (variable quality, different ophthalmologic diseases). The outcome measures were: precision and recall of the extraction of CEC, correctness of CEC density estimation, detection of ungradable images. A classical approach based on grayscale morphology and water shedding was pursued for comparison. There was good agreement between the automated image analysis and the manual annotation from the U-Net. R-square from Pearson's correlation was 0.96. Recall of CEC averaged 0.34 and precision 0.84. The U-Net correctly predicted the CEC density in a large set of images of healthy and diseased corneas, including images of poor quality. It robustly ignored image regions with poor visibility of CEC. The classical approach, however, did not provide acceptable results. R-square from Pearson's correlation with the ground truth was as low as 0.35.