  We report torsional oscillator supersolid studies of highly disordered
samples of solid $^4$He. In an attempt to approach the amorphous or glassy
state of the solid, we prepare our samples by rapid freezing from the normal
phase of liquid $^4$He. Less than two minutes is required for the entire
process of freezing and the subsequent cooling of the sample to below 1 K. The
supersolid signals observed for such samples are remarkably large, exceeding 20
% of the entire solid helium moment of inertia. These results, taken with the
finding that the magnitude of the small supersolid signals observed in our
earlier experiments can be reduced to an unobservable level by annealing,
strongly suggest that the supersolid state exists for the disordered or glassy
state of helium and is absent in high quality crystals of solid $^4$He.
