The task of partially spoofed audio localization aims to accurately determine audio authenticity at a frame level. Although some works have achieved encouraging results, utilizing boundary information within a single model remains an unexplored research topic. In this work, we propose a novel method called Boundary-aware Attention Mechanism (BAM). Specifically, it consists of two core modules: Boundary Enhancement and Boundary Frame-wise Attention. The former assembles the intra-frame and inter-frame information to extract discriminative boundary features that are subsequently used for boundary position detection and authenticity decision, while the latter leverages boundary prediction results to explicitly control the feature interaction between frames, which achieves effective discrimination between real and fake frames. Experimental results on PartialSpoof database demonstrate our proposed method achieves the best performance. The code is available at https://github.com/media-sec-lab/BAM.