Invert the sink upside-down so you have an unobstructed view and access to the bottom rim of the sink. Wipe down the sink underside -- where it would touch the countertop when put into place -- with a damp cloth to remove dust and any debris.
Wipe down the surface of the counter with a damp rag to also remove dust and debris. Focus the cleaning along the countertop surface where the sink's edge or rim would rest once put into place.
Run a thin, long bead of adhesive caulk along the edge of the rim on the upside-down sink. The unbroken bead of caulk needs to be only 3/8-inch thick. Place it 1/4 inch inside the outer edge of the sink's rim.
Lift the sink up carefully by grasping it from the drain hole and/or faucet mount holes. Invert the sink so that it is upright and is ready to be properly set onto the countertop.
Insert and lay the sink into the countertop hole, centering it and orienting the faucet holes toward the back of the vanity. Press the sink down evenly and firmly to push out the bead of adhesive caulk and fuse the sink to the counter.
Wet your finger and run it all along the edge of the sink and countertop to remove any excess caulk that's been pushed out. Wipe off your finger in a wet rag or paper towel to dispose of the excess caulk.