Turn off the water to the toilet using the stop-flow valve on the pipe. Flush the toilet several times until you’ve emptied the tank as much as possible.
Loosen the nut on the pipe that feeds water into the tank. Remove the pipe. Loosen the locknut on the spud valve, using the pipe wrench, until you can lift off the toilet tank. Empty the remaining water from the tank into the bathtub and set the tank upside down.
Twist off the spud valve and its washer by hand.
Clean the area where the spud valve has been seated with a rag to remove all dirt and grease. Allow the area to dry, then screw the new spud valve in place by hand. Place a new washer onto the spud valve, ensuring that the protruding end of the valve faces toward the toilet bowl.
Apply caulk to the washer where it will meet the toilet bowl. Spread the caulk evenly over the washer for extra protection against leakage. Set the tank back on top of the toilet bowl. Tighten the spud valve locknut with the pipe wrench; don’t make it too tight or you can risk stripping the threads.
Reattach the water pipe to the tank, but don’t turn on the water. Let the caulk dry for 24 hours before filling and using the toilet.
Fill the tank by turning on the stop-flow valve. Flush the toilet when the tank is full, and check for any leaks where the tank meets the bowl. If there are leaks, empty the tank and check that the valve is seated correctly.