We study the problem of optimally choosing bearing measurement locations for localizing a stationary target in minimum time. The targets are transmitting radio tags, and bearing measurements are acquired from radio signal strength by a robot carrying a direction-sensitive radio antenna. Actively localizing radio tags has many applications in surveillance, search and rescue, and environmental monitoring. Our work is motivated by the task of monitoring radio-tagged invasive fish using autonomous vehicles. An active localization algorithm is provided in order to locate a target up to the desired uncertainty. The time required to locate the target includes time spent traveling as well as taking measurements. Since bearing measurements inferred from radio signals have an inherent ambiguity associated with them, the proposed algorithm chooses measurements to minimize the effect of ambiguous measurements on the target estimate. We present a closed-form bound on the time required to locate a target using the presented active localization strategy. We also present the first known lower bound on the time required by any active localization algorithm (including the unknown optimal). Finally, we bound the ratio of the upper and lower bounds, showing that the expected cost of our algorithm is within a constant factor of the expected cost of the optimal solution. Robust initialization strategies that are motivated by practical sensing limitations are also provided. Our algorithm is shown to reliably locate radio tags to a desired uncertainty in simulations and multiple field experiments.