Turn off the supply of water to the sink, using the cutoff valves for the water lines. You can usually find the valves located beneath the sink in the sink’s cabinet. Turn on the faucets to run any remaining water out of the system before beginning to remove the sink.
Remove any caulk present around the sink’s fixtures by cutting it into pieces with a utility knife. You can then pull it off with a small putty knife. Use the same process to remove the caulk from the joint that surrounds the edge of the sink between the sink and the countertop tiles. Use a rotary tool with a carbide grout blade to cut away grout from the joint if it’s filled with grout instead of caulk.
Use a basin wrench to unscrew the fixture mounting nuts under the sink at the base of the faucets. Some faucets use locking rings instead. Loosen the rings with the wrench until you can unscrew them from the base of the faucet the rest of the way by hand.
Unscrew the water supply tubes from the connection point with the household water supply, using an adjustable wrench. Remove the tubes, then pull the fixtures from the sink.
Remove the large nut around the drainpipe beneath the sink with the basin wrench, and then pull the pipe away.
Pull the sink up and out of the countertop enclosure. If the sink resists due to a layer of sealant beneath the rim, then apply pressure with a pry bar between the rim and the mounting frame shelf beneath the rim, to break the seal. Set the sink aside for disposal.
Sand off any sealant from the countertop surface and the mounting board on the sides of the countertop enclosure opening, using a piece of medium-grit sandpaper. Wipe any residue clear with a piece of cloth.
Place a bead of silicone caulk along the top of the two wooden mounting bars on either side of the countertop enclosure hole. Lift the replacement tile-in sink and then place into the hole. The rims on the sides of the sink should sit on the mounting bars on both sides of the sink hole, with the top of the sink flush or slightly lower than the tiled surface surrounding the enclosure.
Fill the gap between the sink and the countertop tiles with the caulking, to seal in the sink to prevent leaks around the sink's edges. Smooth the caulk out with a damp cloth so that it is level with the tiles and the sink’s rim. Allow the sealant 30 minutes to cure.
Reconnect the drainpipe, and push the faucet supply lines down through the holes in the new sink before reconnecting them. Tighten the faucet mounting nuts to hold the faucets in place, then place a bead of caulking around the base of the faucets to seal them. Wait another 30 minutes before turning the water supply back on and testing the new sink.