I think the above sentence is an idiom. I'm just guessing that the meaning is (roughly): Once we began to tell a lie, we will try to cover our lie with more lies.
留まることを知らない (as I read it from meaning-book.com/): "don't know when to stop".
But I'm not sure how to translate 1つの嘘から始まった連鎖は. (A chain that started from 1 lie?)
Thank you in advance for the guidance.