We present a framework for direct monocular odometry based on depth prediction from a deep neural network. In contrast with existing methods where depth information is only partially exploited, we formulate a novel depth prediction residual which allows us to incorporate multi-view depth information. In addition, we propose to use a truncated robust cost function which prevents considering inconsistent depth estimations. The photometric and depth-prediction measurements are integrated in a tightly-coupled optimization leading to a scale-aware monocular system which does not accumulate scale drift. We demonstrate the validity of our proposal evaluating it on the KITTI odometry dataset and comparing it with state-of-the-art monocular and stereo SLAM systems. Experiments show that our proposal largely outperforms classic monocular SLAM, being 5 to 9 times more precise, with an accuracy which is closer to that of stereo systems.