Cut through and around the speaker surround with a utility knife, and pull the cone back to expose the spider and lead wires. (The surround is the soft material on the outside edge of the speaker. The spider is a tan colored, mesh-like material at the bottom of the speaker cone.)
Snip the lead wires with a wire cutter, snipping close to the voice coil so the wires may be reused.
Cut through and around the spider with a utility knife, and pull the entire cone and spider straight up to remove. (The spider is connected to the voice coil, which is inserted into the speaker magnet.) Use a slight back and forth motion while pulling, if necessary, to remove it.
Cover the exposed voice coil magnet with strips of masking tape to protect it from falling debris.
Remove the outer gaskets from the basket (the speaker metal that holds it all together), and the spider by inserting the utility underneath the gasket, and cutting around the gasket circumference.
Apply acetone with a squirt bottle to the basket and spider gasket area to soften old adhesive, following with a putty knife and sandpaper to remove the adhesive residue. (Use a rotary tool with a sanding attachment to save time.)
Remove the masking tape from the voice coil magnet, and clean the magnet gap with paper towel dampened with acetone wrapped around a strip of cardboard. Insert the paper towel device into the gap, and rub around its circumference.
Spray the magnet gap with compressed air, and repeat the cleaning process with a strip of masking tape wrapped around another strip of cardboard. (Removing all debris from the magnet gap is essential--the voice coil needs to float freely within the gap.)
Apply adhesive supplied with the recone kit to the spider and basket gasket area. Follow all instructions for setting and application on the adhesive tube, as some adhesives may need to sit before assembling components.
Curl the voice coil centering shim (a strip of supplied cardboard) into a tube, and place it into the voice coil magnet gap.
Slide the recone assembly over the voice coil centering shim, with the old lead wires lining up on the same side as the recone assembly wire terminals.
Press the recone assembly into the speaker, and use your fingers to press the spider and surround into the adhesive area. (Use a small, blunt piece of wood, such as a clothes hanger, to press the surround into the adhesive instead of your fingers, if desired.)
Apply adhesive to the new outer gaskets and set them in place.
Remove the voice coil centering shim, and dry-fit the dust cap over the voice coil, marking the outside circumference with a pencil when it is properly centered. Remove the dust cap.
Apply adhesive around the dust cap pencil mark, and place the dust cap in place again, using a strip of masking tape--sticky side out--as a handle, and apply another bead of adhesive to the outside edges of the dust cover. Place a small circular weight over the dust cover and allow the adhesive to set.
Attach the lead wires to the terminals, and solder them in place. Follow proper soldering procedures for best results; avoid overheating the wires.
Wait for the adhesive set, following the setting time recommendation on the adhesive tube.
Test the speaker.