That was always a problem with me too and the trick is to add a pinch of baking soda to the basket before you brew or two pinches if you like stronger coffee beans. Works really well to neutralize the acidity and there is no after taste just smooth coffee flavor. My friends don't even notice and my friend R. has the same problem only more so. I get the baking soda from a natural food store but I'm sure good old Arm & Hammer is o.k. too.
I've read about a cold brewing process that results in almost 70% less acidity - or so they claim. This creates some kind of coffee extract that hot or cold water can be added to and is supposed to have wonderful flavor and be much easier on the stomach. The machine I saw was called the Hourglass and there's more info about that at their site (they sell them on Amazon too I believe). Someone commented on the original article I found and stated that they used tea bags that you fill yourself and just put coffee in them instead of buying the machine. I haven't had any problems with the regular coffee process or acidity so I haven't tried any of these myself but might provide some relief w/o having to give up the black sludge! Hope that helps.