This paper presents a study using 1D convolutional neural networks (CNNs) for ECG-based authentication. A simple CNN structure is used to both learn the features and do the classification automatically. Two types of CNNs are used in classification as a two-phase process. The "general" CNN is constructed based on global data and used as the preliminary screening, while "person-specific" CNN is constructed using single individual's data and applied as the fine-grained identification. The two-phase identification enables efficient recognition while guarantees a high specificity. Finger ECG signals are collected in different sessions using a mobile device. The proposed algorithm is tested on both within and across session data sets, and on different sample sizes. Results show that the proposed method achieves promising performance in authentication, with a 2.0% EER over 12000 beats. Due to its simple nature, the proposed system is highly applicable for practical application.