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Currently, the diagnosis of Alzheimer's disease is a complex and error-prone process. Improving this diagnosis could allow earlier detection of the disease and improve the quality of life of patients and their families. For this work, we will use 249 brain images from two modalities: PET and MRI, taken from the ADNI database, and labelled into three classes according to the degree of development of Alzheimer's disease. We propose the development of a convolutional neural network to perform the classification of these images, during which, we will study the appropriate depth of the networks for this problem, the importance of pre-processing medical images, the use of transfer learning and data augmentation techniques as tools to reduce the effects of the problem of having too little data, and the simultaneous use of multiple medical imaging modalities. We also propose the application of an evaluation method that guarantees a good degree of repeatability of the results even when using a small dataset. Following this evaluation method, our best final model, which makes use of transfer learning with COVID-19 data, achieves an accuracy d 68\%. In addition, in an independent test set, this same model achieves 70\% accuracy, a promising result given the small size of our dataset. We further conclude that augmenting the depth of the networks helps with this problem, that image pre-processing is a fundamental process to address this type of medical problem, and that the use of data augmentation and the use of pre-trained networks with images of other diseases can provide significant improvements.