We report the Dzyaloshinskii-Moriya interaction (DMI) in Pt/Co/Re films. The DMI has attracted great interest in recent years due to its fundamental role in the stabilization of magnetic skyrmions, which are promising building blocks for next-generation data storage and information processing devices. The DMI at ferromagnetic-metal/heavy-metal (FM/HM) interfaces is of particular practical importance since the DMI can be designed by selecting the materials for the HM and FM layers. In particular, an additive large DMI can be achieved in HM/FM/HM systems when the top and bottom interfaces show the same sign of the interfacial DMI. Here, we investigate the DMI in Pt/Co/Re films to explore the possibility of achieving a large DMI. By measuring the current-induced hysteresis loop shift, we find that the interfacial DMI in the Pt/Co/Re structure is as large as that of Ir/Co/Pt asymmetric multilayers, which is widely known for its large DMI. The large DMI in this system is due to the additional DMI at the Pt/Co and Co/Re interfaces, which is consistent with theoretical predictions. This result provides useful information for designing the size and stabilizing magnetic skyrmions, as well as for understanding the DMI at interfaces.