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Teleosts exhibit extensive diversity of sex determination (SD) systems and mechanisms, providing the opportunity to study the evolution of SD and sex chromosomes. Here we sequenced the genome of the common lumpfish (Cyclopterus lumpus Linnaeus), a species of increasing importance to aquaculture, and identified the SD region and master SD locus using a 70 K single nucleotide polymorphism array and tissue-specific expression data. The chromosome-level assembly identified 25 diploid chromosomes with a total size of 572.89 Mb, a scaffold N50 of 23.86 Mb and genome annotation-predicted 21,480 protein-coding genes. Genome-wide association analysis located a highly sex-associated region on chromosome 13, suggesting that anti-Müllerian hormone (AMH) is the putative SD factor. Linkage disequilibrium and heterozygosity across chromosome 13 support a proto-XX/XY system, with an absence of widespread chromosome divergence between sexes. We identified three copies of AMH in the lumpfish primary and alternate haplotype assemblies localized in the SD region. Comparison to sequences from other teleosts suggested a monophyletic relationship and conservation within the Cottioidei. One AMH copy showed similarity to AMH/AMHY in a related species and was also the only copy with expression in testis tissue, suggesting this copy may be the functional copy of AMH in lumpfish. The two other copies arranged in tandem inverted duplication were highly similar, suggesting a recent duplication event. This study provides a resource for the study of early sex chromosome evolution and novel genomic resources that benefits lumpfish conservation management and aquaculture.