Although lyrics generation has achieved significant progress in recent years, it has limited practical applications because the generated lyrics cannot be performed without composing compatible melodies. In this work, we bridge this practical gap by proposing a song rewriting system which rewrites the lyrics of an existing song such that the generated lyrics are compatible with the rhythm of the existing melody and thus singable. In particular, we propose SongRewriter, a controllable Chinese lyric generation and editing system which assists users without prior knowledge of melody composition. The system is trained by a randomized multi-level masking strategy which produces a unified model for generating entirely new lyrics or editing a few fragments. To improve the controllabiliy of the generation process, we further incorporate a keyword prompt to control the lexical choices of the content and propose novel decoding constraints and a vowel modeling task to enable flexible end and internal rhyme schemes. While prior rhyming metrics are mainly for rap lyrics, we propose three novel rhyming evaluation metrics for song lyrics. Both automatic and human evaluations show that the proposed model performs better than the state-of-the-art models in both contents and rhyming quality. Our code and models implemented in MindSpore Lite tool will be available.