Trim around the hole with a utility knife. Extend the hole so that at least one ceiling joist is exposed. This will make it easier to install the patch drywall. Trim out the hole until it is evenly shaped, preferably in a rectangle.
Measure for the patch piece of drywall with a measuring tape. Get the width and length measurements for the patch.
Cut the new drywall to fit with a utility knife. Score the back side of the panel with a utility knife. Snap it over a table or your knee to break the board. Cut through any paper that does not break away with the utility knife.
Fit the drywall panel into the hole and install it with drywall screws and a cordless screwdriver. Carefully fit the drywall into the bare space. Insert at least three screws through the drywall into the ceiling joist.
Tape over the seams of the patched drywall. Press the self-adhesive drywall tape into the ceiling firmly, especially over the stippled area. If it will not adhere, use the utility knife to scrape away the stippling over where you need to tape.
Apply joint compound over the tape and over your screws. Scrape on a thin layer of compound with the edge of a putty knife over all of the tape and the screw marks. Let this set until it is hard to the touch. This usually takes between four and six hours. Sand it with a fine-grit sandpaper once it is hard. The surface of the patch should be smooth.
Lay drop cloths down underneath the ceiling and over the walls. This will protect them from overspray.
Mix joint compound with a small amount of water and ceiling paint. Place everything in a bucket. Stir the mixture together until it is the consistency of pancake batter. Place the mixture into the hopper of the ceiling texture gun.
Spray texture over the patched area. Apply the first coat thinly. Look at the texture between the newly patched spot and the old ceiling. If necessary, add more texture until these two areas match.