Install a stud wall from the end of the existing wall to the floor and make sure that you put insulation behind the wall. Apply a vapor barrier over the stud wall and then install a drywall over that. At this point you have two choices: You can remove the existing stucco and then apply new stucco to the entire wall, or you can keep the existing stucco and add new stucco to the new section of the wall. If you choose the latter, the stucco may not have a uniform texture and the color of the new stucco may be slightly different from the old. A simple solution to this problem is to install some horizontal wood trim over the line where the old and new walls meet.