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Bruch's membrane (BM) segmentation on optical coherence tomography (OCT) is a pivotal step for the diagnosis and follow-up of age-related macular degeneration (AMD), one of the leading causes of blindness in the developed world. Automated BM segmentation methods exist, but they usually do not account for the anatomical coherence of the results, neither provide feedback on the confidence of the prediction. These factors limit the applicability of these systems in real-world scenarios. With this in mind, we propose an end-to-end deep learning method for automated BM segmentation in AMD patients. An Attention U-Net is trained to output a probability density function of the BM position, while taking into account the natural curvature of the surface. Besides the surface position, the method also estimates an A-scan wise uncertainty measure of the segmentation output. Subsequently, the A-scans with high uncertainty are interpolated using thin plate splines (TPS). We tested our method with ablation studies on an internal dataset with 138 patients covering all three AMD stages, and achieved a mean absolute localization error of 4.10 um. In addition, the proposed segmentation method was compared against the state-of-the-art methods and showed a superior performance on an external publicly available dataset from a different patient cohort and OCT device, demonstrating strong generalization ability.