We present the results of a search for dark matter weakly interacting massive particles (WIMPs) in the mass range below 20 GeV/c^{2} using a target of low-radioactivity argon with a 6786.0 kg d exposure. The data were obtained using the DarkSide-50 apparatus at Laboratori Nazionali del Gran Sasso. The analysis is based on the ionization signal, for which the DarkSide-50 time projection chamber is fully efficient at 0.1 keVee. The observed rate in the detector at 0.5 keVee is about 1.5 event/keVee/kg/d and is almost entirely accounted for by known background sources. We obtain a 90% C.L. exclusion limit above 1.8 GeV/c^{2} for the spin-independent cross section of dark matter WIMPs on nucleons, extending the exclusion region for dark matter below previous limits in the range 1.8-6 GeV/c^{2}.