The problem of craze failure near the tip of a crack embedded inside a craze is investigated by modeling the crazed material as a highly anisotropic network of springs. This model is based on the presence of cross-tie fibrils in the craze microstructure. These cross-tie fibrils give the craze some small lateral load-bearing capacity so that they can transfer stress between the main fibrils. This load transfer mechanism allows the force on the fibril directly ahead of the crack tip in the center of the craze to reach the breaking force of the chain even though the force on a main fibril as it is being drawn at the craze/bulk interface is much lower. When the craze is sufficiently wide, the discrete network model can be approximated as an anisotropic continuum. Explicit expressions are derived which relate the shear and tensile modulus of the crazed material to the underlying microstructural variables such as fibril spacing, fibril diameter and volume fraction. The predictions of the continuum model are compared with those of the discrete model. We focus on the case of a thin craze where the continuum approximation is shown to be inadequate. The results of our model are used to predict the molecular weight dependence of the fracture toughness of polymer glasses, fracture toughness of diluted entanglement networks, and the kinetics of polymer welding.