This paper presents an algorithm for assigning flight departure delays under probabilistic airport capacity. The algorithm dynamically adapts to weather forecasts by revising, if necessary, departure delays. The proposed algorithm leverages state-of-the-art optimization techniques that have appeared in recent literature. As a case study, the algorithm is applied to assigning departure delays to flights scheduled to arrive at San Francisco International Airport in the presence of uncertainty in the fog clearance time. The cumulative distribution function of fog clearance time was estimated from historical data. Using daily weather forecasts to update the probabilities of fog clearance times resulted in improvement of the algorithm's performance. Experimental results also indicate that if the proposed algorithm is applied to assign ground delays to flights inbound at San Francisco International airport, overall delays could be reduced up to 25% compared to current level.