Live streaming demands low latency and small rebuffering time. Existing ABR schemes often fail to meet these requirements because inaccurate bandwidth prediction degrades the performance. In this work, we propose the Frame-level Bandwidth Segmentation and Playback-time based Bandwidth Prediction (FBS-PBP) method. FBS-PBP performs bandwidth sampling at the video frame level, leverages the piecewise stability to enhance network bandwidth feature extraction, and takes the average bandwidth within the playback time of the buffer occupancy as the target to reduce rebuffering time. Simulations demonstrate that FBS-PBP significantly improves bandwidth prediction accuracy. Moreover, FBS-PBP respectively helps ThroughputRule and MPC to reduce the rebuffering ratio by 4% to 13%, and the average latency by 5.26% and 4.11%, while maintaining low latency. With the help of FBS-PBP, MPC performs as good as MERINA in terms of QoE.