The random-loose-packing fraction of uniform spheres at the limit of zero gravitational force is determined to be 0.555±0.005. This structure corresponds to a sphere packing at its rigidity-percolation threshold. The onset of dilatancy was also measured to be at approximately the same packing fraction. Preliminary evidence also indicates that shear thickening of suspensions with macroscopic spheres and a negative pore liquid pressure develops at the rigidity threshold. These provide further evidence that a hard-sphere glass transition could exist in the packing fraction range of 0.555 to 0.645. © 1990 The American Physical Society.