Turn off the water to the sink's faucet by turning both shutoff valves under the sink clockwise by hand. Place a bucket under the sink to catch any water in the drain lines and pipes.
Disconnect the drain lines from the bottom of the faucet using a basin wrench. Disconnect the drainpipe sections using either your hands or a pipe wrench. Cut away at the caulk around the edge of the sink, and then lift the sink out of the old counter and set it aside.
Cut away at the caulk that secures the old countertop to the wall or backsplash, as well as any that secures the countertop to the cabinets. Remove any mounting hardware attached to the underside of the countertop, which secures it to the cabinets, using a screwdriver.
Lift the old countertop off the cabinets and set it aside. Spread silicone caulk on the tops of the cabinets and then lower the new countertop into place. Apply silicone caulk to the edges of the new countertop and the wall or backsplash. Screw in any mounting hardware that came with the new countertop into the inside of the cabinets and the underside of the counter.
Place the sink upside down between two sawhorses and spread silicone caulk onto the bottom edge of the sink. Lower the sink into the cutout in the new countertop and attach the water supply lines to the bottom of the faucet using a basin wrench. Wipe away any caulk that spreads out between the sink and the counter using a rag.