The paper addresses the problem of maintaining the quality of service (QoS) of an orchestration of Web services (WS), which can be affected by exogenous events (i.e., faults). The main challenge in dealing with this problem is that typically the service where a failure is detected is not the one where a fault has occurred: faults have cascade effects on the whole orchestration of services. The paper presents a novel methodology to treat the problem that is not based on Web service (re)composition, but on an adaptive re-execution of the original orchestration. Specifically, an orchestrator Manager exploits an abstract representation of the whole orchestration and a diagnostic module to localize the source of the detected failure. Then, the Manager drives the re-execution of the orchestration by deciding which service activities can be skipped, and which others must be re-executed.