Reconstructing the depth of stereo-endoscopic scenes is an important step in providing accurate guidance in robotic-assisted minimally invasive surgery. Stereo reconstruction has been studied for decades but remains a challenge in endoscopic imaging. Current approaches can easily fail to reconstruct an accurate and smooth 3D model due to textureless tissue appearance in the real surgical scene and occlusion by instruments. To tackle these problems, we propose a dense stereo reconstruction algorithm using convex optimisation with a cost-volume to efficiently and effectively reconstruct a smooth model while maintaining depth discontinuity. The proposed approach has been validated by quantitative evaluation using simulation and real phantom data with known ground truth. We also report qualitative results from real surgical images. The algorithm outperforms state of the art methods and can be easily parallelised to run in real-time on recent graphics hardware.