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This paper introduces a novel approach to speaker-attributed ASR transcription using a neural clustering method. With a parallel processing mechanism, diarisation and ASR can be applied simultaneously, helping to prevent the accumulation of errors from one sub-system to the next in a cascaded system. This is achieved by the use of ASR, trained using a serialised output training method, together with segment-level discriminative neural clustering (SDNC) to assign speaker labels. With SDNC, our system does not require an extra non-neural clustering method to assign speaker labels, thus allowing the entire system to be based on neural networks. Experimental results on the AMI meeting dataset demonstrate that SDNC outperforms spectral clustering (SC) by a 19% relative diarisation error rate (DER) reduction on the AMI Eval set. When compared with the cascaded system with SC, the parallel system with SDNC gives a 7%/4% relative improvement in cpWER on the Dev/Eval set.