Do the laws of physics prohibit counterfactual communication?

It has been conjectured that counterfactual communication is impossible, even for post-selected quantum particles. We strongly challenge this by proposing exactly such a counterfactual scheme where---unambiguously---none of Alices photons that contribute to the message from Bob have been to Bob. We demonstrate counterfactuality experimentally by means of weak measurements, as well as conceptually using consistent histories---thus simultaneously closing both loopholes. Importantly, the fidelity of Alice learning Bobs bit can be made arbitrarily close to unity.