The existing infrared (IR) sea-sky line (SSL) detection algorithms usually treat the SSL as a straight line and, therefore, cannot handle the curved SSL caused by lens distortion and earth curvature. To address this issue, a simple but efficient one-stage end-to-end network termed CSSLNet that is specially designed for curved IR SSL detection is proposed, which treats the SSL as a curve and achieves accurate detection by directly predicting the coefficients of the polynomial. First, by predicting the fitting coefficients, a quadratic polynomial is obtained to describe the curved SSL. Then, to solve the sway problem of the curve, the intersection points of the SSL and image boundaries are used as additional weights to provide an overall constraint of the curve. Finally, to solve the problem of insufficient training samples of the curved SSL, an image augmentation method is employed to enlarge the dataset by performing warping operations on real IR SSL images. The experiments based on the dataset which includes natural straight, natural curve, and artificially curved SSL indicated that the proposed algorithm could achieve precise detection on both straight and curved SSL, which is comparable with the state-of-the-art in precision and speed. Moreover, the CSSLNet is the first algorithm that is specially designed for curved SSL detection, which can be a baseline for future research.