Software-defined networks (SDNs) facilitate more efficient routing of traffic flows using centralized network view. On the other hand, traditional distributed routing still enjoys the advantage of better scalability, robustness, and swift reaction to events such as failure. There are therefore significant potential benefits to adopt a hybrid operation where both distributed and centralized routing mechanisms co-exist. This hybrid operation however imposes a new challenge to network stability since a poor and inconsistent design can lead to repeated route switching when the two control mechanisms take turns to adjust the routes. In this paper, we discuss ways of solving the stability problem. We first define stability for hybrid SDNs and then establish a per-priority stabilizing framework to obtain stable routing patterns. For each priority class, we discuss three approaches to reach hybrid SDN stability: global optimization, greedy, and local search. It is argued that the proposed local search provides the best tradeoff among cost performance, computational complexity, and route disturbance. Furthermore, we design a system on a centralized controller, which utilizes those algorithms to stabilize the network. The design is implemented and extensively tested by simulations using realistic network information, including a trace of the Abilene network and data from a tier-1 Internet service providers backbone network.