  In reionized regions of the Universe, gas can only collapse to form stars in
dark matter (DM) haloes which grow to be sufficiently massive. If star
formation is prevented in the minihalo progenitors of such DM haloes at
redshifts z >~ 20, then these haloes will not be self-enriched with metals and
so may host Population (Pop) III star formation. We estimate an upper limit for
the abundance of Pop III star clusters which thus form in the reionized
Universe, as a function of redshift. Depending on the minimum DM halo mass for
star formation, between of the order of one and of the order of a thousand Pop
III star clusters per square degree may be observable at 2 <~ z <~ 7. Thus,
there may be a sufficient number density of Pop III star clusters for detection
in surveys such as the Deep-Wide Survey (DWS) to be conducted by the James Webb
Space Telescope (JWST). We predict that Pop III clusters formed after
reionization are most likely to be found at z >~ 3 and within ~ 40 arcsec (~ 1
Mpc comoving) of DM haloes with masses of ~ 10^11 M_Sun, the descendants of the
haloes at z ~ 20 which host the first galaxies that begin reionization. If the
stellar initial mass function (IMF) is top-heavy the clusters may have
sufficiently high luminosities in both Ly_alpha and He II lambda1640 to be
detected and for constraints to be placed on the Pop III IMF. While a small
fraction of DM haloes with masses as high as ~10^9 M_Sun at redshifts z <~ 4
are not enriched due to star formation in their progenitors, external metal
enrichment due to galactic winds is likely to preclude Pop III star formation
in a large fraction of otherwise unenriched haloes, perhaps even preventing
star formation in pristine haloes altogether after reionization is complete.
