Trace the cutout onto the counter with a pencil. If you do not have a cutout, place the sink upside down on the counter, position its back 2 inches from the wall and trace around the circumference. Remove the sink from the counter and make another circle three-quarters of an inch smaller than the trace. This will be your cut circle.
Drill a 3/8-inch hole just inside the trace mark. Cut inside of the trace outline with a saber saw. When you cut half of the circumference, stop and put a board across the counter. Place a screw through the board into the cutout portion of the counter. This will keep the cutout from falling down before you finish cutting. Finish cutting the counter with the saber saw.
Set the bathroom sink upside down on the counter with the faucet holes toward you, allowing them to clear the counter. Place a 1/8-inch bead of silicone caulk under the faucet assembly and set it into the top of the sink. Slide the attaching washers and securing nuts to the faucet handle stems. Tighten the nuts with a socket wrench to secure the faucet to the sink.
Ask your helper to hold the sink and move it off the counter. Apply a ½-inch bead of plumbers putty around the circumference of the sink hole in the counter. Flip the sink over and set it inside the counter, with the faucet toward the back wall. Press down firmly on all sides of the sink to set firmly in the plumbers putty.
Apply 1/8-inch bead of plumbers putty under the lip of the new sink drain basket and set it into the sink drain hole. Attach the flat washer and retaining nut to the exposed threads of the drain basket under the sink. Tighten the retaining nut with an adjustable wrench.
Connect the plumbing to the new sink, including the water supply lines and the sink trap. Trim the excess plumbers putty from the top of the sink with a utility knife.
Apply a bead of silicone caulk around the edge of the sink where it meets the counter. Turn the water supply on and enjoy the fruits of your labor.