Shut off the main water supply to the house by turning the main water supply handle in line with the pipe. Open the spigot handle to release pressure in the lines.
Unscrew the screw located in the center of the spigot handle counterclockwise using a Phillips screwdriver.
Remove the spigot handle from the stem by pulling it directly from the stem. Remove the packing nut located behind the handle using an adjustable wrench. (Turn it counterclockwise.) The packing nut will slip right off when it is completely unscrewed.
Unwrap the packing from the end of the stem where the packing nut was just removed. Wrap new packing around the threads of the stem in a single layer. (If too much packing is used the packing nut will not fit correctly.)
Replace the packing nut by screwing it on clockwise with the adjustable wrench. Place the handle back on the end of the stem, and tighten down the center screw with the Phillips screwdriver.
Turn the main water supply back on. Turn the spigot on to fill the lines with water, then shut it off.