In this work, we consider direction-of-arrival (DoA) estimation in the presence of extreme noise using Deep Learning (DL). In particular, we introduce a Convolutional Neural Network (CNN) that predicts angular directions using the sample covariance matrix estimate. The network is trained from multi-channel data of the true array manifold matrix in the low signal-to-noise-ratio (SNR) regime. By adopting an on-grid approach, we model the problem as a multi-label classification task and train the CNN to predict DoAs across all SNRs. The proposed architecture demonstrates enhanced robustness in the presence of noise, and resilience to a relatively small number of snapshots. Moreover, it is able to resolve angles within the grid resolution. Experimental results demonstrate significant performance gains in the low-SNR regime compared to state-of-the-art methods and without the requirement of any parameter tuning in both cases of correlated and uncorrelated sources. Finally, we relax the assumption that the number of sources is known a priori and present a training method, where the CNN learns to infer their number and predict the DoAs with high confidence. The increased robustness of the proposed solution is highly desirable in challenging scenarios that arise in several fields, ranging from wireless array sensors to acoustic microphones or sonars.