Deep neural networks usually require accurate and a large number of annotations to achieve outstanding performance in medical image segmentation. One-shot segmentation and weakly-supervised learning are promising research directions that lower labeling effort by learning a new class from only one annotated image and utilizing coarse labels instead, respectively. Previous works usually fail to leverage the anatomical structure and suffer from class imbalance and low contrast problems. Hence, we present an innovative framework for 3D medical image segmentation with one-shot and weakly-supervised settings. Firstly a propagation-reconstruction network is proposed to project scribbles from annotated volume to unlabeled 3D images based on the assumption that anatomical patterns in different human bodies are similar. Then a dual-level feature denoising module is designed to refine the scribbles based on anatomical- and pixel-level features. After expanding the scribbles to pseudo masks, we could train a segmentation model for the new class with the noisy label training strategy. Experiments on one abdomen and one head-and-neck CT dataset show the proposed method obtains significant improvement over the state-of-the-art methods and performs robustly even under severe class imbalance and low contrast.