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We tackle the problem of minimum-time flight for a quadrotor through a sequence of waypoints in the presence of obstacles while exploiting the full quadrotor dynamics. Early works relied on simplified dynamics or polynomial trajectory representations that did not exploit the full actuator potential of the quadrotor, and, thus, resulted in suboptimal solutions. Recent works can plan minimum-time trajectories; yet, the trajectories are executed with control methods that do not account for obstacles. Thus, a successful execution of such trajectories is prone to errors due to model mismatch and in-flight disturbances. To this end, we leverage deep reinforcement learning and classical topological path planning to train robust neural-network controllers for minimum-time quadrotor flight in cluttered environments. The resulting neural network controller demonstrates substantially better performance of up to 19\% over state-of-the-art methods. More importantly, the learned policy solves the planning and control problem simultaneously online to account for disturbances, thus achieving much higher robustness. As such, the presented method achieves 100% success rate of flying minimum-time policies without collision, while traditional planning and control approaches achieve only 40%. The proposed method is validated in both simulation and the real world, with quadrotor speeds of up to 42km/h and accelerations of 3.6g.