Recent deep learning techniques promise high hopes for self-driving cars while there are still many issues to be addressed such as uncertainties (e.g., extreme weather conditions) in learned models. In this work, for the uncertainty-aware lane keeping, we first propose a convolutional mixture density network (CMDN) model that estimates the lateral position error, the yaw angle error, and their corresponding uncertainties from the camera vision. We then establish a vision-based uncertainty-aware lane keeping strategy in which a high-level reinforcement learning policy hierarchically modulates the reference longitudinal speed as well as the low-level lateral control. Finally, we evaluate the robustness of our strategy against the uncertainties of the learned CMDN model coming from unseen or noisy situations, as compared to the conventional lane keeping strategy without taking into account such uncertainties. Our uncertainty-aware strategy outperformed the conventional lane keeping strategy, without a lane departure in our test scenario during high-uncertainty periods with random occurrences of fog and rain situations on the road. The successfully trained deep reinforcement learning agent slows down the vehicle speed and tries to minimize the lateral error during high uncertainty situations similarly to what human drivers would do in such situations.