Cut out the damaged area of drywall. Measure a square or rectangular area around the stain, and mark it to create a cutting line for the saw. Carefully cut out the area, using a hand or jab saw.
Cut a piece of plywood to fit behind the hole. The piece needs to be wider than the hole by at least 4 inches, and it needs to fit into the hole. Place the piece behind the hole, and attach it to the existing drywall on each side with drywall screws. Use a minimum of four screws.
Use the cut-out piece of drywall to cut a new piece of drywall. Place the cutout on top of the new piece, and cut around it so that the new piece is the same size and shape as the cut-out piece.
Place the new piece in the hole. It will fit tight if it's cut properly. Attach the new piece to the plywood, using drywall screws. Use a drill to push the screws through, holding the new piece in place.
Apply drywall tape around the edges of the new piece, securing it to the old drywall.
Mud around and over the drywall tape with drywall compound. Apply with a putty knife, and allow it to dry. Sand smooth after drying.
Apply a thin layer of plaster over the drywall compound, using a putty knife. Let it dry, and sand smooth.