  We show that a polarized affine variety admits a Ricci flat K\"ahler cone
metric, if and only if it is K-stable. This generalizes Chen-Donaldson-Sun's
solution of the Yau-Tian-Donaldson conjecture to K\"ahler cones, or
equivalently, Sasakian manifolds. As an application we show that the
five-sphere admits infinitely many families of Sasaki-Einstein metrics.
