Place a damp, cotton cloth over the curling corner. Iron over the cloth to straighten the corner underneath using the high, "wool" setting and steam. Turn the rug over and repeat the process.
Mist the curling ends of the rug with water. Place a plastic bag over the wet portion of the rug. Then set heavy objects such as a vase or a stack of telephone books on top of the curling edge. Leave the objects there for 24 hours or more.
Place double-stick tape or double-stick hook-and-loop tape on the underside of the rug in each curling corner. Then stick the corners to the floor.