Place the old axe head into a vise so that the handle sticks out horizontally. Use a hand saw to cut the handle free as close to the head as possible.
Fit 3/8-inch drill bit into a power drill. Drill a hole through the wood handle that remains inside the head. Make several holes. Chisel the wood away. Dig out any remaining debris using a woodworking file or small wire brush.
Scuff the part of the new handle that inserts into the axe head just enough to roughen the surface slightly.
Place the new axe head in the vice so the bottom faces up and the hole is slightly above the top of the vise. Force the handle down into the head by hand. Place a wood block over the end of the new handle and tap with a hammer.
Remove the axe from the vise. Position a wood wedge into the center split at the top of the handle in the axe head. Place the bottom end of the handle on the floor, and lay a piece of scrap wood over the wedge. Hammer on the block lightly to push the wedge into the handle and force the two sides tight against the inside of the axe head.
Use a hand saw to cut the remaining wedge from the head and make it flush with the the handle top. Sand the top with fine-grit sandpaper and slightly round the edges to remove the sharp point.