When a faucet handle will not turn off, the source of the problem is probably a stripped-out stem. A faucet stem or cartridge attaches to the handle and controls the flow of water to the faucet. Hiring a plumber to replace a faucet stem could cost as much as $100, according to 2010 prices. But the job can be completed by the homeowner with a little preparation and the right parts in about 30 minutes.
Stop the flow of water to the faucet by turning off the shut-off valves under the sink or by turning off the main water shut-off for the house.
Remove the button cap at the top of the handle by prying it off with a screwdriver. The button cap usually will have a "C" or an "H" imprinted on it, designating it as the hot-water or cold-water handle. Unscrew the screw at the center of the handle and remove the handle. Some handles may screw off and there may be an Alan screw at the base of the handle that will have to be removed in order to remove the rest of the handle.
Remove the locking nut with an adjustable wrench. The locking nut fits around the stem and holds it in place in the faucet body.
Pluck the washer out of the cavity with needle-nose pliers. Some faucets will not have a washer. The washer is a thin, disc-like ring.
Locate the retaining clip at the base of the stem that holds it in place. The retaining clip is horseshoe-shaped and can be removed by grasping it with needle nose pliers and pulling.
Pull out the faucet stem with a pair of pliers. The stem may take some twisting and pulling, but it will come out.
Smear silicone grease on the new stem and in the handle cavity.
Install the new stem by reversing the process used to remove the old one. Be sure that the hole in the faucet stem is in the six o'clock position when installed. Turn on the water supply before using the faucet.