Adaptive networks, which combine topological evolution of the network with dynamics on the network, are ubiquitous across disciplines. Examples include technical distribution networks such as road networks and the internet, natural and biological networks, and social science networks. These networks often interact with or depend upon other networks, resulting in coupled adaptive networks. In this paper we study susceptible-infected-susceptible (SIS) epidemic dynamics on coupled adaptive networks, where susceptible nodes are able to avoid contact with infected nodes by rewiring their intranetwork connections. However, infected nodes can pass the disease through internetwork connections, which do not change with time: The dependencies between the coupled networks remain constant. We develop an analytical formalism for these systems and validate it using extensive numerical simulation. We find that stability is increased by increasing the number of internetwork links, in the sense that the range of parameters over which both endemic and healthy states coexist (both states are reachable depending on the initial conditions) becomes smaller. Finally, we find a new stable state that does not appear in the case of a single adaptive network but only in the case of weakly coupled networks, in which the infection is endemic in one network but neither becomes endemic nor dies out in the other. Instead, it persists only at the nodes that are coupled to nodes in the other network through internetwork links. We speculate on the implications of these findings.