Attach a 4-by-6 that spans the width of the entire floor to the floor joists using a rafter tie and screws.
Place a 4-by-6 post on the floor with a 6-by-6 post on top directly under the 4-by-6 attached to the floor joists.
Place a 1/4 inch steel plate underneath each floor jack.
Crank the jack so that it is tight to the 4-by-4 piece of wood on the floor joists.
Raise the jack 1/4 inch each day until the floor joists are supported. You should not hear any noise or cracking.
Cut the nails holding the rotted floor joist in place and slide it out from the 4-by-6 brace. You may need to cut it in half to do so.
Cut a new piece of wood that is the same size as the old floor joist using a saw.
Slide the new floor joist in place. Hammer it to the sill plates at each end of the joist to secure it in place using 16d nails.
Lower the jack 1/4 inch each day to release the tension of the jacks slowly. Remove the 4-by-6 from the floor joists.