UAVs are becoming popular in agriculture, however, they usually use time-consuming row-by-row flight paths. This paper presents a deep-reinforcement-learning-based approach for path planning to efficiently localize weeds in agricultural fields using UAVs with minimal flight-path length. The method combines prior knowledge about the field containing uncertain, low-resolution weed locations with in-flight weed detections. The search policy was learned using deep Q-learning. We trained the agent in simulation, allowing a thorough evaluation of the weed distribution, typical errors in the perception system, prior knowledge, and different stopping criteria on the planner's performance. When weeds were non-uniformly distributed over the field, the agent found them faster than a row-by-row path, showing its capability to learn and exploit the weed distribution. Detection errors and prior knowledge quality had a minor effect on the performance, indicating that the learned search policy was robust to detection errors and did not need detailed prior knowledge. The agent also learned to terminate the search. To test the transferability of the learned policy to a real-world scenario, the planner was tested on real-world image data without further training, which showed a 66% shorter path compared to a row-by-row path at the cost of a 10% lower percentage of found weeds. Strengths and weaknesses of the planner for practical application are comprehensively discussed, and directions for further development are provided. Overall, it is concluded that the learned search policy can improve the efficiency of finding non-uniformly distributed weeds using a UAV and shows potential for use in agricultural practice.