Recent rapid growth in the number of satellite-constellation programs for remote sensing and communications, thanks to the availability of small-size and low-cost satellites, provides impetus for high capacity laser communication (lasercom) in space. Quantum communication can enhance the overall performance of lasercom, and also enables intrinsically hack-proof secure communication known as Quantum Key Distribution (QKD). Here, we report a quantum communication experiment between a micro-satellite (48 kg and 50 cm cube) in a low earth orbit and a ground station with single-photon counters. Non-orthogonal polarization states were transmitted from the satellite at a 10-MHz repetition rate. On the ground, by post-processing the received quantum states at an average of 0.14 photons/pulse, clock data recovery and polarization reference-frame synchronization were successfully done even under remarkable Doppler shifts. A quantum bit error rate below 5% was measured, demonstrating the feasibility of quantum communication in a real scenario from space.