The climate state of an ice-covered Neoproterozoic “Snowball Earth” climate is highly sensitive to the presence of dust on the ice. Other authors have noted that the accumulation of dust can lower the albedo of an ice-covered world enough to allow deglaciation even with fairly modest amounts of greenhouse gas warming. Here, we model the accumulation, transport, and removal of dust in a coupled ice-atmosphere-dust climate model, operating within the simplified framework of a one-dimensional energy balance climate model. We confirm that the buildup of low-albedo dust makes our model easier to deglaciate (requiring orders of magnitude less carbon dioxide than a no-dust model) and discover novel climate feedbacks and chaos in the dusty ice system. As dust accumulates, its low albedo warms the ice and can cause it to melt: in our model, the meltwater carries away dust, leading to both a simple negative feedback (the “dust thermostat”) and chaotic oscillations. The feedback is robust over a broad range of model parameters, but the chaotic oscillations are sensitive to model resolution.