Atrial fibrillation (AF) represents the most prevalent type of cardiac arrhythmia for which treatment may require patients to undergo ablation therapy. In this surgery cardiac tissues are locally scarred on purpose to prevent electrical signals from causing arrhythmia. Patient-specific cardiac digital twin models show great potential for personalized ablation therapy, however, they demand accurate semantic segmentation of healthy and scarred tissue typically obtained from late gadolinium enhanced (LGE) magnetic resonance (MR) scans. In this work we propose the Left Atrial Cascading Refinement CNN (LA-CaRe-CNN), which aims to accurately segment the left atrium as well as left atrial scar tissue from LGE MR scans. LA-CaRe-CNN is a 2-stage CNN cascade that is trained end-to-end in 3D, where Stage 1 generates a prediction for the left atrium, which is then refined in Stage 2 in conjunction with the original image information to obtain a prediction for the left atrial scar tissue. To account for domain shift towards domains unknown during training, we employ strong intensity and spatial augmentation to increase the diversity of the training dataset. Our proposed method based on a 5-fold ensemble achieves great segmentation results, namely, 89.21% DSC and 1.6969 mm ASSD for the left atrium, as well as 64.59% DSC and 91.80% G-DSC for the more challenging left atrial scar tissue. Thus, segmentations obtained through LA-CaRe-CNN show great potential for the generation of patient-specific cardiac digital twin models and downstream tasks like personalized targeted ablation therapy to treat AF.