A novel criterion for brittle fracture of entangled polymer liquids is presented: Crack initiation follows from rupture of primary C-C bonds, when the strain energy of an entanglement segment reaches the energy of the covalent bond. Thermal fluctuations lead to a short-time concentration of the strain energy on one C-C bond of the entanglement segment, and the chain ruptures. This limits the maximum achievable stretch of entanglement segments to a critical stretch of fc≤6. Recent experimental data of Huang et al. [Phys. Rev. Lett. 117, 087801 (2016)]] and Huang and Hassager [Soft Matter 13, 3470-3474 (2017)] on fracture of solutions of nearly monodisperse polystyrenes dissolved in oligomeric styrene and of a well characterized polydisperse polystyrene melt, are in general agreement with this fracture criterion. For quantitative agreement, finite extensibility effects have to be considered.