This paper describes the design and the performance of the timing detector developed by the TOTEM Collaboration for the Roman Pots (RPs) to measure the Time-Of-Flight (TOF) of the protons produced in central diffractive interactions at the LHC. The measurement of the TOF of the protons allows the determination of the longitudinal position of the proton interaction vertex and its association with one of the vertices reconstructed by the CMS detectors. The TOF detector is based on single crystal Chemical Vapor Deposition (scCVD) diamond plates and is designed to measure the protons TOF with about 50 ps time precision. This upgrade to the TOTEM apparatus will be used in the LHC run 2 and will tag the central diffractive events up to an interac...