A possible Dark Matter model postulates that it interacts with Standard Model particles only through a massive photon-like vector particle, called dark photon or A′. The PADME experiment at the DAΦNE Beam-Test Facility (BTF) in Frascati is designed to detect dark photons produced in positron on fixed target annihilations decaying to dark matter (e+e-→γA′) by measuring the final state missing mass. The DAQ system of the PADME experiment will handle a total of 921 channels, with a DAQ rate of 50 Hz. All channels will be acquired using the CAEN V1742 board, a 32 channels 5 GS/s digitizer based on the DRS4 chip. Two such boards were successfully used during the 2015 and 2016 tests at the BTF, where a complete DAQ system, prototypal to the one which will be used for the final experiment, was set up. In this paper we will report on the details of the DAQ system, with specific reference to our experience with the V1742 board.