Topology optimization (TO) has developed rapidly recently. However, topology optimization with stress constraints still faces many challenges due to its highly non-linear properties which will cause inefficient computation, iterative oscillation, and convergence guarantee problems. At the same time, isogeometric analysis (IGA) is accepted by more and more researchers, and it has become one important tool in the field of topology optimization because of its high fidelity. In this paper, we focus on topology optimization with stress constraints based on isogeometric analysis to improve computation efficiency and stability. A new hybrid solver combining the alternating direction method of multipliers and the method of moving asymptotes (ADMM-MMA) is proposed to solve this problem. We first generate an initial feasible point by alternating direction method of multipliers (ADMM) in virtue of the rapid initial descent property. After that, we adopt the method of moving asymptotes (MMA) to get the final results. Several benchmark examples are used to verify the proposed method, and the results show its feasibility and effectiveness.