In the cold dark matter model of structure formation, galaxies are assembled hierarchically from mergers and the accretion of subclumps. This process is expected to leave residual substructure in the Galactic dark halo, including partially disrupted clumps and their associated tidal debris. We develop a model for such halo substructure and study its implications for dark matter (WIMP and axion) detection experiments. We combine the Press-Schechter model for the distribution of halo subclump masses with N-body simulations of the evolution and disruption of individual clumps as they orbit through the evolving Galaxy to derive the probability that the Earth is passing through a subclump or stream of a given density. Our results suggest that it is likely that the local complement of dark matter particles includes a 1-5% contribution from a single clump. The implications for dark matter detection experiments are significant, since the disrupted clump is composed of a `cold flow of high-velocity particles. We describe the distinctive features due to halo clumps that would be seen in the energy and angular spectra of detection experiments. The annual modulation of these features would have a different signature and phase from that for a smooth halo and, in principle, would allow one to discern the direction of motion of the clump relative to the Galactic center.
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