Sex chromosomes vary greatly in their age and levels of differentiation across the tree of life. This variation is largely due to the rates of sex chromosome turnover in different lineages; however, we still lack an explanation for why sex chromosomes are so conserved in some lineages (e.g. mammals, birds) but so labile in others (e.g. teleosts, amphibians). To identify general mechanisms driving transitions in sex determination systems or forces which favour their conservation, we first require empirical data on sex chromosome systems from multiple lineages. Stickleback fishes are a valuable model lineage for the study of sex chromosome evolution due to variation in sex chromosome systems between closely-related species. Here, we identify the sex chromosome and a strong candidate for the master sex determination gene in the brook stickleback, Culaea inconstans. Using whole-genome sequencing of wild-caught samples and a lab cross, we identify AmhY, a male specific duplication of the gene Amh, as the candidate master sex determination gene. AmhY resides on Chromosome 20 in C. inconstans and is likely a recent duplication, as both AmhY and the sex-linked region of Chromosome 20 show little sequence divergence. Importantly, this duplicate AmhY represents the second independent duplication and recruitment of Amh as the sex determination gene in stickleback and the eighth example known across teleosts. We discuss this convergence in the context of sex chromosome turnovers and the role that the Amh/AmhrII pathway, which is crucial for sex determination, may play in the evolution of sex chromosomes in teleosts.