  The direct detection of sub-GeV dark matter interacting with nucleons is
hampered by the low recoil energies induced by scatterings in the detectors.
This experimental difficulty is avoided in the scenario of boosted dark matter
where a component of dark matter particles is endowed with large kinetic
energies. In this Letter, we point out that the current evaporation of
primordial black holes with masses from $10^{14}$ to $10^{16}$ g is a source of
boosted light dark matter with energies of tens to hundreds of MeV. Focusing on
the XENON1T experiment, we show that these relativistic dark matter particles
could give rise to a signal orders of magnitude larger than the present upper
bounds. Therefore, we are able to significantly constrain the combined
parameter space of primordial black holes and sub-GeV dark matter. In the
presence of primordial black holes with a mass of $10^{15}~\mathrm{g}$ and an
abundance compatible with present bounds, the limits on DM-nucleon
cross-section are improved by four orders of magnitude.
