When dealing with overlapped speech, the performance of automatic speech recognition (ASR) systems substantially degrades as they are designed for single-talker speech. To enhance ASR performance in conversational or meeting environments, continuous speaker separation (CSS) is commonly employed. However, CSS requires a short separation window to avoid many speakers inside the window and sequential grouping of discontinuous speech segments. To address these limitations, we introduce a new multi-channel framework called "speaker separation via neural diarization" (SSND) for meeting environments. Our approach utilizes an end-to-end diarization system to identify the speech activity of each individual speaker. By leveraging estimated speaker boundaries, we generate a sequence of embeddings, which in turn facilitate the assignment of speakers to the outputs of a multi-talker separation model. SSND addresses the permutation ambiguity issue of talker-independent speaker separation during the diarization phase through location-based training, rather than during the separation process. This unique approach allows multiple non-overlapped speakers to be assigned to the same output stream, making it possible to efficiently process long segments-a task impossible with CSS. Additionally, SSND is naturally suitable for speaker-attributed ASR. We evaluate our proposed diarization and separation methods on the open LibriCSS dataset, advancing state-of-the-art diarization and ASR results by a large margin.