Recently, the duals of Federer’s curvature measures, called dual curvature measures, were discovered by Huang, Lutwak, Yang & Zhang (ACTA, 2016). In the same paper, they posed the dual Minkowski problem, the characterization problem for dual curvature measures, and proved existence results when the index, q, is in (0,n). The dual Minkowski problem includes the Aleksandrov problem (q = 0) and the logarithmic Minkowski problem (q = n) as special cases. In the current work, a complete solution to the dual Minkowski problem whenever q < 0, including both existence and uniqueness, is presented.