Evaluating the residual stress in welds is great important, and X-ray stress measurements are superior to other stress measurement methods because they are nondestructive and provide high-resolution results. However, it is well known that texture in welds is found in some materials such as austenitic stainless steel, and that such texture can greatly reduce the measurement accuracy. Although much knowledge regarding the application of X-ray stress measurement to materials with texture has been accumulated, most studies have considered texture under uniaxial stress. However, weld residual stress is typically multi-axial stress. In this study, an attempt was made to improve the accuracy of X-ray residual-stress measurements for welds with texture. To evaluate the stress in welds with texture under multi-axial stress, a new method was developed for evaluating plane stress from the elastic compliance, the value fraction of the crystal, and measurements in two orthogonal directions. A new method was also developed for evaluating the value fraction of the crystal in welds with texture. Moreover, in a SUS316L weld specimen, the residual stress evaluated by the proposed method agreed very well with that evaluated by the stress relief method. Thus, the proposed method has the potential to be effective for measuring residual stress in welds with texture.