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We use the replica method to study the ideal glass transition of a liquid of identical hard spheres. We obtain estimates of the configurational entropy in the liquid phase, of the Kauzmann packing fraction φ K⁠, in the range of 0.58–0.62, and of the random close packing density φ c⁠, in the range of 0.64–0.67, depending on the approximation we use for the equation of state of the liquid. We also compute the pair-correlation function in the glassy states (ie, dense amorphous packings) and we find that the mean coordination number at φ c is equal to 6. All these results compare well with numerical simulations and with other existing theories.