Our ever-aging society faces the growing problem of neurodegenerative diseases, in particular dementia. Magnetic Resonance Imaging provides a unique tool for non-invasive investigation of these brain diseases. However, it is extremely difficult for neurologists to identify complex disease patterns from large amounts of three-dimensional images. In contrast, machine learning excels at automatic pattern recognition from large amounts of data. In particular, deep learning has achieved impressive results in image classification. Unfortunately, its application to medical image classification remains difficult. We consider two reasons for this difficulty: First, volumetric medical image data is considerably scarcer than natural images. Second, the complexity of 3D medical images is much higher compared to common 2D images. To address the problem of small data set size, we assemble the largest dataset ever used for training a deep 3D convolutional neural network to classify brain images as healthy (HC), mild cognitive impairment (MCI) or Alzheimers disease (AD). We use more than 20.000 images from subjects of these three classes, which is almost 9x the size of the previously largest data set. The problem of high dimensionality is addressed by using a deep 3D convolutional neural network, which is state-of-the-art in large-scale image classification. We exploit its ability to process the images directly, only with standard preprocessing, but without the need for elaborate feature engineering. Compared to other work, our workflow is considerably simpler, which increases clinical applicability. Accuracy is measured on the ADNI+AIBL data sets, and the independent CADDementia benchmark.