We extend the Quantum Cluster Equilibrium (QCE/3-21G-level) theory of liquid water to include larger icelike clusters, including tetrahedral and fullerenelike clusters up to 26-mers. A low-energy tetrakaidecahedral 24-mer (labeled w24_0, with two hexagonal apical faces bridged by 12 pentagons in a cooperatively charge-balanced H-bond pattern) is found to lead to a new low-T phase that bounds both liquid and vapor regions in first-order transition lines, giving rise to a true QCE triple point. We characterize the microstructural composition and macroscopic properties of this "Bucky-ice" phase and examine its dependence on calculated energetics and vibrational properties of constituent clusters. Although the Bucky-ice phase differs in significant respects from physical ice-I (e.g., melting point similar to 20 K too high, molar volume similar to 5% too low), it manifests qualitatively correct thermodynamic features of true ice polymorphs, suggesting an important role of clathratelike clusters in the liquid/solid transition region.