Automatic Playlist Continuation (APC) refers to the task of automatically adding one or more tracks to a playlist that match the user preferences and the song characteristics of the original playlist. In previous approaches, recommendations have often relied on dot product to account for inter-music similarities, but this approach has struggled to convey important inequality properties. To address these issues, as metric learning itself can close the distance between similar samples, we propose a novel Metric Learning based Two-stage Music Recommendation (MLTMR) for Automatic Playlist Continuation. Specifically, in order to filter out tracks related to user preferences and playlist topic, first, we compute a recommendation candidate set using Mahalanobis distance, and apply Adversarial Personalized Ranking (APR) to enhance the robustness of the recommendation model. Second, we use the Siamese Neural Networks (SNN) to further recommend tracks that are similar to the member songs on existing playlists, thereby improving the accuracy of the recommendation. The experimental results illustrate that our proposed model outperforms six state-of-the-art models in two large real-world datasets.