Dark matter pervades the Solar System, free-streaming at the halo's local Galactic orbital velocity and density. As these objects pass through the Solar system, they perturb gravitationally, and thus very weakly, all nearby inertial masses. Making use of this, we propose an approach to the direct detection of dark matter at previously inaccessible intermediate masses (1e14 – 1e20 gm). Such mass scales are relevant for example for dark matter made of primordial black holes or clumped matter in a sequestered sector. If such dark matter exists it will be unambiguously detectable through its inelastic gravitational interaction with the proposed Laser Interferometer Space Antenna (LISA) experiment. We demonstrate the efficacy of this approach by studying the dark matter signal in numerical simulations of the LISA data stream. A more conservative approach – to detect dark matter in the differential acceleration power spectrum – significantly underestimates the expected rates for LISA. Interestingly, while the space-density of 1e15 gm DM objects would be comparable to the space-density of asteroids of similar masses, such ``light matter’’ contaminants are readily detectable in reflected Solar light, allowing for the elimination of the major background.