As far as I know, this is false. The cycle stops repeating when one has reached enlightenment; to my knowledge, what happens after that point is irrelevant - one can't go back from enlightenment, and certainly not due to the body decaying (which is a major topic of Buddhist meditation anyway).
Is there any Pali canon source (I'm only interested in Theravada) that backs up the concept of no decay = no rebirth?
This idea, known in the text as annihilationism, was rejected by the Buddha as far as I know.