Entity normalization is an important information extraction task which has gained renewed attention in the last decade, particularly in the biomedical and life science domains. In these domains, and more generally in all specialized domains, this task is still challenging for the latest machine learning-based approaches, which have difficulty handling highly multi-class and few-shot learning problems. To address this issue, we propose C-Norm, a new neural approach which synergistically combines standard and weak supervision, ontological knowledge integration and distributional semantics.