The transmission and reception of polarized quantum-limited signals from space is of capital interest for a variety of fundamental-physics experiments and quantum-communication protocols. Specifically, Quantum Key Distribution (QKD) deals with the problem of distributing unconditionally-secure cryptographic keys between two parties. Enabling this technology from space is a critical step for developing a truly-secure global communication network. The National Institute of Information and Communications Technology (NICT, Japan) performed the first successful measurement on the ground of a quantum-limited signal from a satellite in experiments carried out on early August in 2016. The SOTA (Small Optical TrAnsponder) lasercom terminal onboard the LEO satellite SOCRATES (Space Optical Communications Research Advanced Technology Satellite) was utilized for this purpose. Two non-orthogonally polarized signals in the ~800-nm band and modulated at 10 MHz were transmitted by SOTA and received in the single-photon regime by using a 1-m Cassegrain telescope on a ground station located in an urban area of Tokyo (Japan). In these experiments, after compensating the Doppler effect induced by the fast motion of the satellite, a QKD-enabling QBER (Quantum Bit Error Rate) below 5% was measured with estimated key rates in the order of several Kbit/s, proving the feasibility of quantum communications in a real scenario from space for the first time.