Remove the existing window. Remove the interior trim from the bottom edge of the window by carefully tapping it away from the window sill with a hammer and pry bar. The nails holding the trim will gradually pull out if you take your time. This will expose the entire window sill.
Cut the center of the window sill with a circular saw. Set the saw to a shallow depth so you don't cut into the wall beneath the sill. When the sill is cut in the middle, lever the two sides out with a pry bar or large screwdriver. Clean out any wood debris from the area from where you've removed the sill.
Cut a new sill to the correct dimensions, with a jig saw, using wood of the same thickness. Paint the sill with an exterior house paint.
Place the sill into the window opening from the front and gently tap it into position with a hammer. Nail the sill to the wood underneath with 6d finishing nails. Caulk around the perimeter of the sill with acrylic weatherproof caulk.