Multi-modal journey planning, which allows multiple types of transport within a single trip, is becoming increasingly popular, due to a strong practical interest and an increasing availability of data. In real life, transport networks feature uncertainty. Yet, most approaches assume a deterministic environment, making plans more prone to failures such as major delays in the arrival. We model the scenario as a non-deterministic planning problem with continuous time and time-dependent probabilities of non-deterministic effects. We present new hardness results. We introduce a heuristic search planner, based on Weighted AO* (WAO*). The planner includes search enhancements such as sound pruning, based on state dominance, and an admissible heuristic. Focusing on plans that are robust to uncertainty, we demonstrate our ideas on data compiled from real historical data from Dublin, Ireland. Repeated calls to WAO*, with decreasing weights, have a good any-time performance. Our search enhancements play an important role in the planner's performance. Copyright © 2013, Association for the Advancement of Artificial Intelligence. All rights reserved.