/ AbstractWe report on a search for sub-GeV dark matter (DM) particles interacting with electrons using the DAMIC-M prototype detector at the Modane Underground Laboratory. The data feature a significantly lower detector single $e^-$ rate (factor 50) compared to our previous search, while also accumulating a ten times larger exposure of $\sim$1.3 kg-day. DM interactions in the skipper charge-coupled devices (CCDs) are searched for as patterns of two or three consecutive pixels with a total charge between 2 and 4 $e^-$. We find 144 candidates of 2 $e^-$ and 1 candidate of 4 $e^-$, where 141.5 and 0.071, respectively, are expected from background. With no evidence of a DM signal, we place stringent constraints on DM particles with masses between 1 and 1000 MeV/$c^2$ interacting with electrons through an ultra-light or heavy mediator. For large ranges of DM masses below 1 GeV/c$^2$, we exclude theoretically-motivated benchmark scenarios where hidden-sector particles are produced as a major component of DM in the Universe through the freeze-in or freeze-out mechanisms.