Cut a 3/4-inch-thick plywood board to dimensions one-half-inch shorter than the length and width of the door frame. Cover with a thin layer of wood glue. Press an identical board on top of the glued surface, putting the two least attractive surfaces together. Clamp the boards firmly together and allow to dry overnight. Remove the clamps and set them aside.
Use a drill to remove the old door from its hinges and uninstall the doorknob and latch hardware from the old door. Pay special attention to the order of steps with which the hardware is uninstalled, writing the steps down or taking photographs if necessary to facilitate an easier reinstallation in the new door. Position the new door in its intended spot and mark the location of the hinge screw holes. Drill out holes in each marking, being careful not to go through to the other side of the door. Attach the door to the hinges using 1 1/4-inch wood screws.
Swing the new door closed and mark the location where the latch should be installed so it lines up with the closing bracket already installed in the door frame. Drill out the marked area in the end of the new door so that it matches the location of the drilled-out space in the door frame. Make a hole through the side of the new door as well, in the same position as the hole on the side of the old door.
Insert the doorknob and latch hardware into the new door and tighten it in place, reversing the steps with which the hardware was removed from the old door. Swing the door shut and test how well the latch lines up with the frame. Adjust the position of the frame bracket if necessary, as that adjustment is easier than adjusting the position of the new doorknob.