  Primordial black holes (PBHs) are of fundamental interest in cosmology and
astrophysics, and have received much attention as a dark matter candidate and
as a potential source of gravitational waves. One possible PBH formation
mechanism is the gravitational collapse of cosmic strings. Thus far, the
entirety of the literature on PBH production from cosmic strings has focused on
the collapse of (quasi)circular cosmic string loops, which make up only a tiny
fraction of the cosmic loop population. We demonstrate here a novel PBH
formation mechanism: the collapse of a small segment of cosmic string in the
neighbourhood of a cusp. Using the hoop conjecture, we show that collapse is
inevitable whenever a cusp appears on a macroscopically-large loop, forming a
PBH whose rest mass is smaller than the mass of the loop by a factor of the
dimensionless string tension squared, $(G\mu)^2$. Since cusps are generic
features of cosmic string loops, and do not rely on finely-tuned loop
configurations like circular collapse, this implies that cosmic strings produce
PBHs in far greater numbers than has previously been recognised. The resulting
PBHs are highly spinning and boosted to ultrarelativistic velocities; they
populate a unique region of the BH mass-spin parameter space, and are therefore
a "smoking gun" observational signature of cosmic strings. We derive new
constraints on $G\mu$ from the evaporation of cusp-collapse PBHs, and update
existing constraints on $G\mu$ from gravitational-wave searches.
