  Dark matter elastic scattering off nuclei can result in the excitation and
ionization of the recoiling atom through the so-called Migdal effect. The
energy deposition from the ionization electron adds to the energy deposited by
the recoiling nuclear system and allows for the detection of interactions of
sub-GeV/c$^2$ mass dark matter. We present new constraints for sub-GeV/c$^2$
dark matter using the dual-phase liquid argon time projection chamber of the
DarkSide-50 experiment with an exposure of (12306 $\pm$ 184) kg d. The analysis
is based on the ionization signal alone and significantly enhances the
sensitivity of DarkSide-50, enabling sensitivity to dark matter with masses
down to 40 MeV/c$^2$. Furthermore, it sets the most stringent upper limit on
the spin independent dark matter nucleon cross section for masses below $3.6$
GeV/c$^2$.
