Twist the bottom cap of the toothbrush counter-clockwise 45 degrees, or until it doesn't twist anymore. Grab the bottom cap and the body of the toothbrush and pull them apart. Set the outer shell of the toothbrush aside.
Position the toothbrush so it sits upright. Locate the two thin pieces of wire on the circuit board that are near the bottom of the toothbrush. Sever these wires with a soldering iron as close as you can to the circuit board. All it takes is a slight tap of the hot soldering iron to cut the wire. The bottom piece can now be removed from the rest of the toothbrush.
Squeeze both sides of the bottom piece of the toothbrush. There are two hooks that keep it connected, but they will come loose once you squeeze them. Pull the bottom piece out and set it aside. This piece holds a spring and coil mechanism that is necessary for recharging a battery, so do not lose them or let them fall out.
Look at the top side of the circuit board. There are two thin pieces of metal that connect the circuit board to another part of the toothbrush. Sever these pieces carefully with the soldering iron. Pull the circuit board away from the toothbrush to reveal the battery.
Push the battery out through the bottom of the toothbrush. Put the new battery in with the positive side facing up toward the top.
Put the circuit board back on the toothbrush as it was before. Locate the two thin pieces of metal that you had to sever in order to remove the circuit board. Heat these up with the soldering iron and then hold the soldering flux against them. If the metal is hot enough, the flux will melt to reconnect the metal pieces.
Put the bottom piece of the toothbrush back in place. The underside of this piece should have a small notch. Line this notch up with the center of the circuit board and push the piece in until it clicks.
Solder the two thin wires back to the circuit board. Be sure they go in the same spots that they originally were.
Put the outer shell of the toothbrush back on and twist it clockwise to lock it in place.