Did the Buddha believe in rebirth and reincarnation?

The Buddha didn't believe in a Supreme Being, or a universal soul. The Buddha didn't even believe in a soul. Then how could he preach rebirth and reincarnation when he didn't even believe in a soul? What the Theravada tradition says about this? I know that the Theravada tradition is closest to the original teachings of Buddha, hence I would like to know what it says about this.