We experimentally investigate discrete avalanches of grains, driven by a low inflow rate, on an erodible pile in a channel. We observe intermittency between one regime, in which avalanches are quasiperiodic and system spanning, and another, in which they pass at irregular intervals and have a power-law size distribution. Observations are robust to changes of inflow rate and grain type and require no tuning of external parameters. We demonstrate that the state of the pile's surface determines whether avalanche fronts propagate to the end of the channel or stop partway down, and we introduce a toy model for the latter case that reproduces the observed power-law size distribution. We suggest direct applications to avalanches of pharmaceutical and geophysical grains, and the possibility of reconciling the "self-organized criticality" predicted by several authors with the hysteretic behavior described by others.