  It has been conjectured that counterfactual communication is impossible, even
for post-selected quantum particles. We strongly challenge this by proposing
precisely such a counterfactual scheme where -- unambiguously -- none of
Alice's photons that correctly contribute to her information about Bob's
message have been to Bob. We demonstrate counterfactuality experimentally by
means of weak measurements, and conceptually using consistent histories -- thus
simultaneously satisfying both criteria without loopholes. Importantly, the
fidelity of Alice learning Bob's bit can be made arbitrarily close to unity.
