We consider the power-efficient design of an Internet protocol (IP)-over-wavelength division multiplexing (WDM) network, tackling the problem of installing optical and IP layer equipment to satisfy traffic requirements. We take into account routing constraints and consider a comprehensive set of realistic scenarios defined by network topology, traffic matrix, and power consumption values of network devices in both layers. Furthermore, besides defining and solving an optimal integer linear programming model, we propose an efficient heuristic to solve the problem up to medium-sized networks. The proposed heuristic requires at most 30% of additional power with respect to the optimal solution, but with a significantly reduced complexity. We show that the largest power consumption is due to line cards and routers rather than WDM equipment. Furthermore, we find that multipath routing reduces the network power consumption with respect to single-path routing, but not significantly. Finally, we show that a two-step design procedure, in which power is separately minimized in each layer (i.e., IP and WDM layers), can find results with a total power consumption comparable to the one achieved by a more complex joint multilayer design procedure.