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We report neutron-scattering measurements on highly supercooled liquid gallium droplets showing a vanishing quasi-elastic frequency width as the temperature is reduced. The width tends towards zero at a temperature within 1 standard deviation of the value ${T}_{L}=150\ifmmode\pm\else\textpm\fi{}1$ K at which gallium is always observed to become crystalline. This suggests the interpretation of ${T}_{L}$ as a metastability limit, below which the liquid state becomes unstable with respect to infinitesimal fluctuations.