This paper presents the HRVConformer, a novel deep learning architecture for the classification of hypoxic-ischemic encephalopathy (HIE) using the instantaneous heart rate (HR) signal. Unlike conventional approaches that rely on handcrafted features, HRVConformer directly processes raw HR signals in an end-to-end manner, capturing both local and long-range dependencies through a hybrid Convolution-Transformer framework. By integrating convolutional layers for local feature extraction and Transformer-based attention mechanisms for global context modelling, the architecture effectively enhances signal representation and classification performance. The model was trained using supervised learning on a large HR dataset consisting of 1,573 one-hour epochs, including 259 one-hour expert-annotated epochs and a substantial set of weakly labelled data. A 314-hour validation set provided a robust performance estimation, while an independent 215-hour dataset with expert annotations was reserved for final testing. HR signals were extracted from electrocardiogram (ECG) recordings using an improved Pan-Tompkins algorithm, which significantly enhanced both signal quality and data availability. Experimental results demonstrate that the HRVConformer achieves an AUC of 83.23\% and accuracy of 74.56\% on the test set. These results surpass the performance of the Transformer, ResNet50 and fully convolutional networks baselines, highlighting the advantages of integrating convolutional and Transformer-based components for HR-based HIE classification. The proposed method provides a promising step toward a more accurate and automated assessment of HIE using HR signals. The code is available at: https://github.com/syu-kylin/HRVConformer.