If the neutral component of weak-scale dark matter is accompanied by a charged excitation separated by a mass gap of less than ~20 MeV, weakly interacting massive particles (WIMPs) can form stable bound states with nuclei. We show that the recent progress in experiments searching for neutrinoless double-beta decay sets the first direct constraint on the exoergic reaction of WIMP-nucleus bound state formation. We calculate the rate for such a process in representative models and show that the double-beta decay experiments provide unique sensitivity to a large fraction of parameter space of the WIMP doublet model, complementary to constraints imposed by cosmology and direct collider searches.