Exploitation of Best Effort Distributed Computing Infrastructures (BE-DCIs) allow operators to maximize the utilization of the infrastructures, and users to access the unused resources at relatively low cost. Because providers do not guarantee that the computing resources remain available to the user during the entire execution of their applications, they offer a diminished Quality of Service (QoS) compared to traditional infrastructures. Profiling the execution of Bag-of-Tasks (BoT) applications on several kinds of BE-DCIs demonstrates that their task completion rate drops near the end of the execution. In this report, we present the SpeQuloS service which enhances the QoS of BoT applications executed on BE-DCIs by reducing the execution time, improving its stability, and reporting to users a predicted completion time. SpeQuloS monitors the execution of the BoT on the BE-DCIs, and dynamically supplies fast and reliable Cloud resources when the critical part of the BoT is executed. We present the design and development of the framework and several strategies to decide when and how Cloud resources should be provisioned. Performance evaluation using simulations shows that SpeQuloS fulfill its objectives. It speeds-up the execution of BoTs, in the best cases by a factor greater than 2, while offloading less than 2.5% of the workload to the Cloud. We report on preliminary results after a complex deployment as part of the European Desktop Grid Infrastructure.