Song Sun-Cheon boxed at the 1956 and 1960 Olympics, winning a silver medal as a bantamweight in 1956. He lost the final to German Wolfgang Behrendt, which was disputed by Korean boxing officials. Song also won a bronze medal at the 1958 Asian Games as a featherweight.
Song later became an educator, teaching at Cheongju University in North Chungcheong Province and Yongin University in Gyeonggi Province. After his retirement he was made a professor emeritus. He also was a founding member of the Korean Olympic Medalists’ Association and served as chairman.