Background sound is an informative form of art that is helpful in providing a more immersive experience in real-application voice conversion (VC) scenarios. However, prior research about VC, mainly focusing on clean voices, pay rare attention to VC with background sound. The critical problem for preserving background sound in VC is inevitable speech distortion by the neural separation model and the cascade mismatch between the source separation model and the VC model. In this paper, we propose an end-to-end framework via multi-task learning which sequentially cascades a source separation (SS) module, a bottleneck feature extraction module and a VC module. Specifically, the source separation task explicitly considers critical phase information and confines the distortion caused by the imperfect separation process. The source separation task, the typical VC task and the unified task shares a uniform reconstruction loss constrained by joint training to reduce the mismatch between the SS and VC modules. Experimental results demonstrate that our proposed framework significantly outperforms the baseline systems while achieving comparable quality and speaker similarity to the VC models trained with clean data.