The aim of this paper is to implement a system to facilitate the diagnosis of multiple sclerosis (MS) in its initial stages. It does so using a convolutional neural network (CNN) to classify images captured with swept-source optical coherence tomography (SS-OCT). SS-OCT images from 48 control subjects and 48 recently diagnosed MS patients have been used. These images show the thicknesses (45 × 60 points) of the following structures: complete retina, retinal nerve fiber layer, two ganglion cell layers (GCL+, GCL++) and choroid. The Cohen distance is used to identify the structures and the regions within them with greatest discriminant capacity. The original database of OCT images is augmented by a deep convolutional generative adversarial network to expand the CNN's training set. The retinal structures with greatest discriminant capacity are the GCL++ (44.99% of image points), complete retina (26.71%) and GCL+ (22.93%). Thresholding these images and using them as inputs to a CNN comprising two convolution modules and one classification module obtains sensitivity = specificity = 1.0. Feature pre-selection and the use of a convolutional neural network may be a promising, nonharmful, low-cost, easy-to-perform and effective means of assisting the early diagnosis of MS based on SS-OCT thickness data.