Are you saying you were accepted off their waitlist yesterday, or were you already accepted and just now got the $ offer? If the latter, you must have been holding onto at least 2 acceptances on the 15th, indicating to me that you were going to have a dilemma come today no matter what Michigan offered you on the 15th. Care to clarify?
All opinion, but: I don't think your other school is going to drop you without warning on Monday morning. Next week, I assume that all schools will be told which students are holding multiple acceptances, and I assume from that schools will get in touch with you to force your hand.
So I think you have some time, but do not bury your head in the sand. Be proactive - contact the other school and explain that you need more time - be specific if you can and give a date in the next week by when you will notify them.