Segmentation of cardiac structures is one of the fundamental steps to estimate volumetric indices of the heart. This step is still performed semiautomatically in clinical routine and is, thus, prone to interobserver and intraobserver variabilities. Recent studies have shown that deep learning has the potential to perform fully automatic segmentation. However, the current best solutions still suffer from a lack of robustness in terms of accuracy and number of outliers. The goal of this work is to introduce a novel network designed to improve the overall segmentation accuracy of left ventricular structures (endocardial and epicardial borders) while enhancing the estimation of the corresponding clinical indices and reducing the number of outliers. This network is based on a multistage framework where both the localization and segmentation steps are optimized jointly through an end-to-end scheme. Results obtained on a large open access data set show that our method outperforms the current best-performing deep learning solution with a lighter architecture and achieved an overall segmentation accuracy lower than the intraobserver variability for the epicardial border (i.e., on average a mean absolute error of 1.5 mm and a Hausdorff distance of 5.1mm) with 11% of outliers. Moreover, we demonstrate that our method can closely reproduce the expert analysis for the end-diastolic and end-systolic left ventricular volumes, with a mean correlation of 0.96 and a mean absolute error of 7.6 ml. Concerning the ejection fraction of the left ventricle, results are more contrasted with a mean correlation coefficient of 0.83 and an absolute mean error of 5.0%, producing scores that are slightly below the intraobserver margin. Based on this observation, areas for improvement are suggested.