A 3D model of the human iris provides an additional degree of freedom in iris recognition, which could help identify people in larger databases, even when only a piece of the iris is available. Previously, we reported developing a 3D iris scanner that uses 2D images of the iris from multiple perspectives to reconstruct a 3D model of the iris. This paper focuses on the development of a 3D iris scanner from a single image by means of a Convolutional Neural Network (CNN). The method is based on a depth-estimation CNN for the 3D iris model. A dataset of 26,520 real iris images from 120 subjects, and a dataset of 72,000 synthetic iris images with their aligned depthmaps were created. With these datasets, we trained and compared the depth estimation capabilities of available CNN architectures. We analyzed the performance of our method to estimate the iris depth in multiple ways: using real step pyramid printed 3D models, comparing the results to those of a test set of synthetic images, comparing the results to those of the OCT scans from both eyes of one subject, and generating the 3D rubber sheet from the 3D iris model proving the correspondence with the resulting 2D rubber sheet and binary codes. On a preliminary test the proposed 3D rubber sheet model increased iris recognition performance by 48% with respect to the standard 2D iris code. Other contributions include assessing the scanning resolution, reducing the acquisition and processing time to produce the 3D iris model, and reducing the complexity of the image acquisition system.