This isn't so much plans for a particular paper airplane, as a concept and technique that could be used for a whole new style of airplane, not TOO much more complex or expensive than the typical all-paper airplane.
Step 1: Weighing Paper
Step 2: Weighing Other Things.
One of these surprisingly rigid when inflated, at least compared to a piece of paper. You can attach wings and a tail with tape, weight the nose, and create a number of interesting plane designs. And it's still cheap and still easy to experiment with!
Here we have some balloons, weight in at 1.6 to 2.1 grams. Slightly more when inflated, of course (thereby demonstrating that air has mass!) That may not seem like a lot less than the paper when you're looking at the un-inflated balloon, but that 2.1g balloon is about five feet long when inflated. That's either a very large plane, or ... we can use PART of the balloon!