Turn off the water to the house, then place a towel under the old sink to catch any water and debris. Unfasten the water and drain lines with a plumber's wrench.
Score the edges of the old sink with a utility knife, being careful not to cut into the counter, to loosen the old sink from the counter top.
Pry the edges up with a putty knife and pull the old sink out of the counter cavity. Scrape away any grime and the old caulk from the cavity with a putty knife to get a smooth, clean surface. Clean around the edges of the cavity with water and soap or TSP cleaner. Dry with a cloth.
Set the new sink into the counter cavity to make certain it fits. Then, pull it back out.
Spread a bead of silicone caulk around the rim of the counter top cavity with a caulk gun. Before the caulk hardens, seat the replacement sink into the cavity. Crawl under the counter and attach the water and drain lines. Snug them firmly with a plumber's wrench.
Return to the top of the counter and wipe away with a rag the excess silicone caulk that squeezed from under the sink's rim while you were tightening the lines. Turn the water supply back on.