As long as ACME is owned by Albertson's you should have no problem. We have a Jewel/Osco Preferred card and were able to use it without any problem at the Albertson's on Orange Blossom Trail.
DW wasn't going to ask about the card, but I said what the heck. I had it with me, and sure enough the check said, "Absolutely, we accept all of the Albertson's family of stores cards."
Brian