  We present a new near-field cosmological probe of the initial mass function
(IMF) of the first stars. Specifically, we constrain the lower-mass limit of
the Population III (Pop III) IMF with the total number of stars in large,
unbiased surveys of the Milky Way. We model the early star formation history in
a Milky Way-like halo with a semi-analytic approach, based on Monte-Carlo
sampling of dark matter merger trees, combined with a treatment of the most
important feedback mechanisms. Assuming a logarithmically flat Pop III IMF and
varying its low mass limit, we derive the number of expected survivors of these
first stars, using them to estimate the probability to detect any such Pop III
fossil in stellar archaeological surveys. Following our analysis, the most
promising region to find possible Pop III survivors is the stellar halo of the
Milky Way, which is the best target for future surveys. We find that if no
genuine Pop III survivor is detected in a sample size of $4 \times 10^6$ ($2
\times 10^7$) halo stars with well-controlled selection effects, then we can
exclude the hypothesis that the primordial IMF extended down below $0.8
M_\odot$ at a confidence level of 68% (99%). With the sample size of the
Hamburg/ESO survey, we can tentatively exclude Pop III stars with masses below
$0.65 M_\odot$ with a confidence level of 95%, although this is subject to
significant uncertainties. To fully harness the potential of our approach,
future large surveys are needed that employ uniform, unbiased selection
strategies for high-resolution spectroscopic follow-up.
