The fracture strength under quasi-static steady-state crack growth in an elastic-plastic material joined by a laser weld is analyzed. Laser welding gives high mismatch between the yield stress within the weld and the yield stress in the base material. This is due to the fast termic cycle, which the material undergoes in welding. The elastic properties, on the other hand, are insensitive to the termic cycle, and are therefore essentially the same in the weld and in the base material. The material is described by $J_2$-flow theory, and the analysis is performed by using a numerical algorithm, in which the finite element mesh remains fixed relative to the tip of the growing crack. Fracture is modelled using two different local crack growth criteria. One is a crack opening displacement criterion, while the other is a model in which a cohesive zone is imposed in front of the crack tip along the fracture zone. Both models predict that in general a thinner laser weld gives higher interface strength. Furthermore, both fracture criteria show, that the preferred path of the crack is close outside the weld material; a phenomenon also observed in experiments.