The combination of mobile edge computing (MEC) and blockchain is transforming the current computing services in Internet of Things networks, by offering task offloading solutions with security enhancement enabled by blockchain mining. Nevertheless, these important enabling technologies have been studied separately in most existing works. This article proposes a novel cooperative task offloading and block mining (TOBM) scheme to optimize the system utility in blockchain-empowered MEC. Herein, each edge device (ED) not only handles data tasks but also deals with block mining which makes the system design and optimization highly complex. Therefore, we develop a novel cooperative deep reinforcement learning (DRL) approach which allows EDs to cooperatively offload their data tasks to the MEC server and perform block mining based on a Proof-of-Reputation consensus mechanism. Simulation results demonstrate that the proposed scheme significantly improves offloading utility, reduces blockchain mining latency, and achieves better system utility, compared to other non-cooperative and cooperative schemes.