3D visual grounding aims to localize the target object in a 3D point cloud by a free-form language description. Typically, the sentences describing the target object tend to provide information about its relative relation between other objects and its position within the whole scene. In this work, we propose a relation-aware one-stage framework, named 3D Relative Position-aware Network (3DRP-Net), which can effectively capture the relative spatial relationships between objects and enhance object attributes. Specifically, 1) we propose a 3D Relative Position Multi-head Attention (3DRP-MA) module to analyze relative relations from different directions in the context of object pairs, which helps the model to focus on the specific object relations mentioned in the sentence. 2) We designed a soft-labeling strategy to alleviate the spatial ambiguity caused by redundant points, which further stabilizes and enhances the learning process through a constant and discriminative distribution. Extensive experiments conducted on three benchmarks (i.e., ScanRefer and Nr3D/Sr3D) demonstrate that our method outperforms all the state-of-the-art methods in general.