Lesion volume segmentation in medical imaging is an effective tool for assessing lesion/tumor sizes and monitoring changes in growth. Since manually segmentation of lesion volume is not only time-consuming but also requires radiological experience, current practices rely on an imprecise surrogate called response evaluation criteria in solid tumors (RECIST). Although RECIST measurement is coarse compared with voxel-level annotation, it can reflect the lesion's location, length, and width, resulting in a possibility of segmenting lesion volume directly via RECIST measurement. In this study, a novel weakly-supervised method called RECISTSup is proposed to automatically segment lesion volume via RECIST measurement. Based on RECIST measurement, a new RECIST measurement propagation algorithm is proposed to generate pseudo masks, which are then used to train the segmentation networks. Due to the spatial prior knowledge provided by RECIST measurement, two new losses are also designed to make full use of it. In addition, the automatically segmented lesion results are used to supervise the model training iteratively for further improving segmentation performance. A series of experiments are carried out on three datasets to evaluate the proposed method, including ablation experiments, comparison of various methods, annotation cost analyses, visualization of results. Experimental results show that the proposed RECISTSup achieves the state-of-the-art result compared with other weakly-supervised methods. The results also demonstrate that RECIST measurement can produce similar performance to voxel-level annotation while significantly saving the annotation cost.