This study investigates robust speaker localization for con-tinuous speech separation and speaker diarization, where we use speaker directions to group non-contiguous segments of the same speaker. Assuming that speakers do not move and are located in different directions, the direction of arrival (DOA) information provides an informative cue for accurate sequential grouping and speaker diarization. Our system is block-online in the following sense. Given a block of frames with at most two speakers, we apply a two-speaker separa-tion model to separate (and enhance) the speakers, estimate the DOA of each separated speaker, and group the separation results across blocks based on the DOA estimates. Speaker diarization and speaker-attributed speech recognition results on the LibriCSS corpus demonstrate the effectiveness of the proposed algorithm.