The Palm Sunday Compromise was a bill which passed the US Senate by a vote of 3-0, 97 not present. How is it possible for the Senate to conduct business with the overwhelming majority of its members absent? Where else has this procedure been used? For what reasons does this possibility exist?
The Wikipedia article answers your question.
A majority of Senators (i.e., 51 of the 100) is required to obtain a quorum, and only three senators out of 100 were present when the bill was voted upon. However, the Senate (and the House) conduct their respective businesses under the presumption that a quorum is always present, unless or until a completed quorum call or roll-call vote demonstrates otherwise (e.g., a roll-call vote or quorum call in the Senate failing to get 51 total votes or replies).
Since the 3 senators never questioned a quorum, one was present by definition even though there obviously wasn't.
This is allowed, because it is part of the rules adopted by the senate, their quorum procedures can be found here. Article one section five grants the Senate and House the ability to make their own rules.
Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behaviour, and, with the Concurrence of two thirds, expel a Member.
This may seem counter intuitive, but the supreme court has made rulings that both houses can decide how to determine a quorum in U.S. vs Ballin so the chances of this ever changing is unlikely without a specific amendment or a fairly major change in judicial precedent.