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Early diagnosis, playing an important role in preventing progress and treating the Alzheimer's disease (AD), is based on classification of features extracted from brain images. The features have to accurately capture main AD-related variations of anatomical brain structures, such as, e.g., ventricles size, hippocampus shape, cortical thickness, and brain volume. This paper proposes to predict the AD with a deep 3D convolutional neural network (3D-CNN), which can learn generic features capturing AD biomarkers and adapt to different domain datasets. The 3D-CNN is built upon a 3D convolutional autoencoder, which is pre-trained to capture anatomical shape variations in structural brain MRI scans. Fully connected upper layers of the 3D-CNN are then fine-tuned for each task-specific AD classification. Experiments on the \emph{ADNI} MRI dataset with no skull-stripping preprocessing have shown our 3D-CNN outperforms several conventional classifiers by accuracy and robustness. Abilities of the 3D-CNN to generalize the features learnt and adapt to other domains have been validated on the \emph{CADDementia} dataset.