We present a deep neural network-based method to perform high-precision, robust and real-time 6 DOF visual servoing. The paper describes how to create a dataset simulating various perturbations (occlusions and lighting conditions) from a single real-world image of the scene. A convolutional neural network is fine-tuned using this dataset to estimate the relative pose between two images of the same scene. The output of the network is then employed in a visual servoing control scheme. The method converges robustly even in difficult real-world settings with strong lighting variations and occlusions. A positioning error of less than one millimeter is obtained in experiments with a 6 DOF robot.