Computational stereo has reached a high level of accuracy, but degrades in the presence of occlusions, repeated textures, and correspondence errors along edges. We present a novel approach based on neural networks for depth estimation that combines stereo from dual cameras with stereo from a dual-pixel sensor, which is increasingly common on consumer cameras. Our network uses a novel architecture to fuse these two sources of information and can overcome the above-mentioned limitations of pure binocular stereo matching. Our method provides a dense depth map with sharp edges, which is crucial for computational photography applications like synthetic shallow-depth-of-field or 3D Photos. Additionally, we avoid the inherent ambiguity due to the aperture problem in stereo cameras by designing the stereo baseline to be orthogonal to the dual-pixel baseline. We present experiments and comparisons with state-of-the-art approaches to show that our method offers a substantial improvement over previous works.