Robots have become a key technology in the manufacturing domain by replacing manual labour, while speeding up, increasing accuracy, and reducing the costs of production processes. Nevertheless, unexpected faults can happen anytime during production, potentially leading to component or system failure, resulting in lower productivity, increased costs, and additional manual maintenance burdens. In this context, Digital Twins (DTs) represent a promising technology that aims to improve the performance and reduce robot downtime. DTs leverage the integration of real robot and simulated data during run-time to monitor the robots and deploy mitigating actions when faults are detected. In this paper, we explore the development of a DT for the Universal Robots UR3e robotic arm, seeking to enhance its capabilities, allowing it to autonomously adapt when faults are detected during the execution of pick-and-place tasks. A proof-of-concept is presented, leveraging the principles of self-adaptation through the Monitor-Analyse-Plan-Execute over a Knowledge base (MAPE-K) loop. Results show that the proposed DT framework effectively detects and mitigates faults in real time, reducing task execution interruptions and improving robotic autonomy. These findings highlight the potential of DTs for enhancing fault tolerance and autonomy in robotic manipulators.