Sonic hedgehog (Shh) controls critical cellular decisions between distinct fates in many systems, particularly in stem cells. The Shh network functions as a genetic switch, and we have theoretically and computationally analyzed how its structure can endow it with the ability to switch fate choices at a threshold Shh concentration. The network is composed of a positive transcriptional feedback loop embedded within a negative signaling feedback loop. Specifically, positive feedback by the transcription factor Gli, which upregulates its own expression, leads to a switch that can adopt two distinct states as a function of Shh. However, Gli also upregulates the signaling repressor Patched, negative feedback that reins in the strong Gli autoregulatory loop. Mutations that have been associated with cancer are predicted to yield an irreversible switch to a high Gli state. Finally, stochastic simulation reveals the negative Patched feedback loop serves a critical function of dampening Gli fluctuations to reduce spontaneous state switching and preserve the network's robust, switch-like behavior. Tightly linked positive and negative feedback loops are present in many signaling systems, and the Shh system is therefore likely representative of a large set of gene regulation networks that control stem cell fate throughout development and into adulthood.