We study the unmanned aerial vehicle (UAV) route planning problem for information collection missions performed in terrains with stochastic attributes. Uncertainty is associated with the availability of information, the effectiveness of the search and collection sensors the UAV carries, and the flight time required to travel between target regions in the mission terrain. Additionally, uncertainties in flight duration vary the detection threat exposed in missions performed in nonfriendly terrains. We develop a mixed integer programming model to maximize the expected information collection while limiting the risks of not completing the mission on time and of being detected and restricting the variance imposed on flight duration. The model allows multiple path alternatives between target pairs and revisits to the same target regions. Computational experiments are performed on randomly generated instances to investigate the impact of problem parameters and mission restrictions. We also develop a case study with a military and a civilian application, each of which with different specifications. In addition, we validate that the actual performance of the optimal solution of the model is close to the result reported by the solver via a simulation study. We conclude that the developed model is robust and can be used for practical-sized missions, and the failure rate of its optimal routes in actual missions is negligibly small.