Original concept by Svenbjorn of Aggelgorod, heavily modified by Jari and edited by Alric
Start with a piece of 1" thick polyethylene foam (available from McMaster-Carr - part #9334K25 - or www.foambymail.com) and a good core. I suggest either schedule 80 PVC or fiberglass rods.
Cut a hole in the foam so it fits snugly over the core. Wrap the top of the core with cloth tape (hockey or athletic tape) and glue the foam on with gorilla glue or one of the cheaper versions of the same kind of glue. Make sure you wet the surfaces first. Go VERY heavy on the glue. Extra can be cut off later with an exacto knife. Clamp it together and let it dry. Duct tape is good for holding these together while it dries; pull it off afterwards if you do use it.
Take some strapping tape (easy to find at any Walmart, in the packing tape section - 3m/Scotch is the best brand) and wrap it around the axe head and core (see the picture). It adds a little bit more stability to the axe. Wrap it tight!
Now carefully cut a piece of leather to wrap around the blade to within an inch the current edge of the blade. I use a lighter leather, slightly under belt (4-8 oz) grade for this. Cut a strip of armor grade leather (10 oz +) for the underside. Drill/cut a hole in this leather and slide it onto the core under the axe head. Again, make sure you put it back from the front edge of the blade by about an inch. Use more gorilla glue to attach all the leather.
Put a layer of .5 inch EVA (or blue camping pad) foam across the top, then one layer of coleman camp pad (or blue foam, if you're cheap) across the bottom. Then wrap the entire head, including the foam you just added, in the same .5 inch EVA (or blue camping pad). It keeps anyone from feeling the leather on the sides and getting hurt. This will result in a 2 inch wide blade.
Now add a strip of .5 inch EVA (of blue camping pad) to the blade. I usually add two.
Now, depending on the size of the axes, glue a piece of couch foam - or even better, a high quality open cell like Unifoam - to the front, using 3m Super 77 spray glue. You can skip this step if you're making a small, single-handed axe.
Attach your pommel.
Wrap the haft in blue foam. I have tried using pipe insulation here, and it breaks down to quickly. Another way to do this is to put a strip of EVA along the striking side and wrap the rest in blue foam; this keeps you from having to replace the blue foam as quickly as it wears out from people hitting the haft and or the haft hitting weapons/shields.
All done!
Jari, with his two axes. Picture by Gronk, 2008.