Model experiments are reported on the build-up of granular piles in two dimensions. These show that as the initial density of falling grains is increased, the resulting pile has decreasing final density and its coordination number approaches the low value predicted for the theoretical marginal rigidity state. This provides the first direct experimental evidence for this state of granular matter. We trace the decrease in the coordination number to the dynamics within an advancing yield front between the consolidated pile and the falling grains. We show that the front's size increases with initial density, diverging as the marginal rigidity state is approached.