  Primordial black holes (PBHs) are compact objects proposed to have formed in
the early Universe from the collapse of small-scale over-densities. Their
existence may be detected from the observation of gravitational waves (GWs)
emitted by PBH mergers, if the signals can be distinguished from those produced
by the merging of astrophysical black holes. In this work, we forecast the
capability of the Einstein Telescope, a proposed third-generation GW
observatory, to identify and measure the abundance of a subdominant population
of distant PBHs, using the difference in the redshift evolution of the merger
rate of the two populations as our discriminant. We carefully model the merger
rates and generate realistic mock catalogues of the luminosity distances and
errors that would be obtained from GW signals observed by the Einstein
Telescope. We use two independent statistical methods to analyse the mock data,
finding that, with our more powerful, likelihood-based method, PBH abundances
as small as $f_\mathrm{PBH} \approx 7 \times 10^{-6}$ ($f_\mathrm{PBH} \approx
2\times10^{-6}$) would be distinguishable from $f_\mathrm{PBH} = 0$ at the
level of $3\sigma$ with a one year (ten year) observing run of the Einstein
Telescope. Our mock data generation code, darksirens, is fast, easily
extendable and publicly available on GitLab.
