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Abstract The possible existence of primordial black holes in the stellar-mass window has received considerable attention because their mergers may contribute to current and future gravitational-wave detections. Primordial black hole mergers, together with mergers of black holes originating from Population III stars, are expected to dominate at high redshifts ( z ≳ 10). However, the primordial black hole merger rate density is expected to rise monotonically with redshift, while Population III mergers can only occur after the birth of the first stars. Next-generation gravitational-wave detectors such as the Cosmic Explorer (CE) and Einstein Telescope (ET) can access this distinctive feature in the merger rates as functions of redshift, allowing for direct measurement of the abundance of the two populations and hence for robust constraints on the abundance of primordial black holes. We simulate four months’ worth of data observed by a CE-ET detector network and perform hierarchical Bayesian analysis to recover the merger rate densities. We find that if the universe has no primordial black holes with masses of  ( 10 M ⊙ ) , the projected upper limit on their abundance f PBH as a fraction of dark matter energy density may be as low as f PBHmore »∼  ( 10 − 5 ) , about two orders of magnitude lower than the current upper limits in this mass range. If instead f PBH ≳ 10 −4 , future gravitational-wave observations would exclude f PBH = 0 at the 95% credible interval.« less