Sleep apnea (SA) detection method based on traditional machine learning needs a lot of efforts in feature engineering and classifier design. We constructed a one-dimensional convolutional neural network (CNN) model, which consists in four convolution layers, four pooling layers, two full connection layers and one classification layer. The automatic feature extraction and classification were realized by the structure of the proposed CNN model. The model was verified by the whole night single-channel sleep electrocardiogram (ECG) signals of 70 subjects from the Apnea-ECG dataset. Our results showed that the accuracy of per-segment SA detection was ranged from 80.1% to 88.0%, using the input signals of single-channel ECG signal, RR interval (RRI) sequence, R peak sequence and RRI sequence + R peak sequence respectively. These results indicated that the proposed CNN model was effective and can automatically extract and classify features from the original single-channel ECG signal or its derived signal RRI and R peak sequence. When the input signals were RRI sequence + R peak sequence, the CNN model achieved the best performance. The accuracy, sensitivity and specificity of per-segment SA detection were 88.0%, 85.1% and 89.9%, respectively. And the accuracy of per-recording SA diagnosis was 100%. These findings indicated that the proposed method can effectively improve the accuracy and robustness of SA detection and outperform the methods reported in recent years. The proposed CNN model can be applied to portable screening diagnosis equipment for SA with remote server.