  Primordial black holes in the asteroid-mass window ($\sim 10^{-16}$ to
$10^{-11} \rm M_{\odot}$), which might constitute all the dark matter, can be
captured by stars when they traverse them at low enough velocity. After being
placed on a bound orbit during star formation, they can repeatedly cross the
star if the orbit happens to be highly eccentric, slow down by dynamical
friction and end up in the stellar core. The rate of these captures is highest
in halos of high dark matter density and low velocity dispersion, when the
first stars form at redshift $z \sim 20$. We compute this capture rate for
low-metallicity stars of $0.3$ to $1\rm M_{\odot}$, and find that a high
fraction of these stars formed in the first dwarf galaxies would capture a
primordial black hole, which would then grow by accretion up to a mass that may
be close to the total star mass. We show the capture rate of primordial black
holes does not depend on their mass over this asteroid-mass window, and should
not be much affected by external tidal perturbations. These low-mass stellar
black holes could be discovered today in low-metallicity, old binary systems in
the Milky Way containing a surviving low-mass main-sequence star or a white
dwarf, or via gravitational waves emitted in a merger with another compact
object. No mechanisms in standard stellar evolution theory are known to form
black holes of less than a Chandrasekhar mass, so detecting a low-mass black
hole would fundamentally impact our understanding of stellar evolution, dark
matter and the early Universe.
