The temperature-dependent viscosity η(T) is measured for the equilibrium liquid of the chalcogenide Ag-In-Sb-Te (AIST), the first time this has been reported for a material of actual interest for phase-change memory. The measurements, in the range 829-1254 K, are made using an oscillating-crucible viscometer, and show a liquid with high fragility and low viscosity, similar to liquid pure metals. Combining the high-temperature viscosity measurements with values inferred from crystal growth rates in the supercooled liquid allows the form of η(T) to be estimated over the entire temperature range from above the melting point down to the glass transition. It is then clear that η(T) for liquid AIST cannot be described with a single fragility value, unlike other phase-change chalcogenides such as liquid Ge-Sb-Te. There is clear evidence for a fragile-to-strong crossover on cooling liquid AIST, similar to that analyzed in Te85Ge15. The change in fragility associated with the crossover in both these cases is rather weak, giving a broad temperature range over which η(T) is near-Arrhenius. We discuss how such behavior may be beneficial for the performance of phase-change memory. Consideration of the fragile-to-strong crossover in liquid chalcogenides may be important in tuning compositions to optimize the device performance.