Remove drywall or paneling from the inside of the wall to expose the wall studs and other framing elements. Attach a metal hurricane tie plate to the top plate of the wall and each stud to create a strong connection. Inspect the bottom sill and mark the ends of the rotted section with chalk, at least 12 inches into good material on each side.
Set one bottle jack with a metal lift column, or 4-by-4 treated post on top, to fit between the jack and the bottom of the wall's top plate. Set the jack on a thick piece of lumber, such as a 2-by-10, to cushion the concrete from the force of the lift.
Raise the two jacks one pump at a time until the bottom of the studs separate slightly from the wall sill running along its bottom. When the separation is about 1/4 inch high, stop jacking and lock the jacks in place to support the wall.
Use a reciprocating saw with a wrecking blade to cut the nails in the ends of each stud to separate the sill from the studs. Cut the nails on every stud that is over the rotted section of the sill.
Cut through the sill with the saw at both ends where you marked it, so the rotted section can be removed. Lift up on the sill with a pry bar. Cut any sill bolts that attach it to the slab with the saw. Pry the cut section of the sill out and away from the wall until it is clear of all studs. Lift and dispose of the rotten lumber.
Measure and cut a new piece to fit from treated lumber using a circular saw. Lay it on the floor flat on one wide face, directly in front of the studs. Tap the new sill into place with a hammer until its inside edge is flush with the inside edges of the studs.
Lower the jacks at each end of the wall until the columns or posts can be removed. Drive nails on an angle through the end of each stud into the new sill board.