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We 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 10 times larger exposure of $\sim 1.3\mathrm{kg}\text{−}\mathrm{day}$. DM interactions in the skipper charge-coupled devices (CCDs) are searched for as groups of two or three adjacent pixels with a total charge between 2 and $4e^{-}$. We find 144 candidates of $2e^{-}$and 1 candidate of $4e^{-}$, 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 $1000\mathrm{MeV}/c^2$interacting with electrons through an ultralight or heavy mediator. For large ranges of DM masses below $1\mathrm{GeV}/c^2$, 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.