I have a paperback that I bought new and only read once, but the cover has come unglued from the binding. Does anyone out there know if and how I can put it back together?

i believe your public library can tell you the best way to make the repair

That's a fairly easy one - you just have to be careful not to use too much glue.  Take a serrated steak knife, and rough up the adhesive strip on the binding so that whatever glue you're using will penetrate into it.  Just apply the glue to the outer edge of the spine.  Place the cover on there and get it properly aligned, and then slip folded wax paper sheets between the cover and pages, making sure they lay flat and come right up the where the cover and edge of the page meet.  Wipe away any excess glue.  Wrap the book in a sheet of copier or printer paper and then secure it with large, wide rubber bands - two across the width & evenly spaced, and two up & down the length.  Put the book on a flat, hard surface, and then place weights on top of it.  Bricks work, or you can use a large hardcover book, and then put something heavy like a cast iron skillet on top of that.  Let it dry for at least 6 hours under the weight, and then carefully unwrap it and remove the wax paper, and check to see if the cover bonded well.  I usually use Brodart Bindart Adhesive, which you can buy cheap from ebay, but I've see people say they've used hot melt glue (very carefully applied), and shoe repair adhesive - I think it's called "Shoe Goo" both to good effect.  Somone even said somewhere that they held a hot iron (with a thin dishtowel between them) against the binding, and were able to soften or melt it enough that it reattatched to the cover when it was placed over the hot binding.  Never tried this myself.

Here are some links - one to book repair tips I posted here a while back: http://www.paperbackswap.com/forum/topic.php?t=107768

And one to Brodart's online book repair PDF manual: http://www.ioba.org/StepByStepBookRepair.pdf

I believe they show a cover being reattatched in there.  I think they cut away some of the old adhesive before applying the new binding adhesive to the spine.