In order to ensure the robust actuation of a plan, execution must be adaptable to unexpected situations in the world and to exogenous events. This is critical in domains in which committing to a wrong ordering of actions can cause the plan failure, even when all the actions succeed. We propose an approach to the execution of a task plan that permits some adaptability to unexpected observations of the state while maintaining the validity of the plan through online reasoning. Our approach computes an adaptable, partially-ordered plan from a given totally-ordered plan. The partially-ordered plan is adaptable in that it can exploit beneficial differences between the world and what was expected. The approach is general in that it can be used with any task planner that produces either a totally or a partially-ordered plan. We propose a plan execution algorithm that computes online the complete set of valid totally-ordered plans described by an adaptable partially-ordered plan together with the probability of success for each of them. This set is then used to choose the next action to execute.