Cut a hole in the goose-down pillow. Reach in and remove the ticking with your hand. Place it in a plastic bag.
Hold the fabric you're using for the new ticking against the old ticking. Cut the fabric to the same size as the old ticking. Repeat to get two pieces. Discard the old ticking.
Pin the two pieces of ticking together about 1 inch down from the top of one of the long sides.
Place the material under the needle in the sewing machine so there is about a 1/4-inch seam above where the stitches will go.
Stitch the entire length of the pillowcase. Repeat on the other length and one of the sides, using pins each time to keep the two pieces straight.
Pin three-quarters of the last side that hasn't been stitched. Stitch it with your machine.
Iron the new stitches on the ticking with high heat.
Turn the entire thing inside out through the hole in the ticking.
Stuff the goose down into the new ticking.
Stitch the remaining hole in the ticking with a needle and thread. Tie off the end with several knots to secure it.