  We adapted the Appleby-Battye-Starobinsky model of $F(R)$ gravity towards
describing double cosmological inflation and formation of primordial black
holes with masses up to $10^{19}$ g in the single-field model. We found that it
is possible to get an enhancement of the power spectrum of scalar curvature
perturbations to the level beyond the Hawking (black hole evaporation) limit of
$10^{15}$ g, so that the primordial black holes resulting from gravitational
collapse of those large primordial perturbations can survive in the present
universe and form part of cold dark matter. Our results agree with the current
measurements of cosmic microwave background radiation within $3\sigma$ but
require fine-tuning of the parameters.
