Go to the basement or other area where you can access the bad joist from underneath. Use your hammer and wood chisel to remove soft or crumbling wood from the surface, so there is just solid wood left.
Measure the length of the joist, including the ends where it may go into a cutout in the walls. Cut the new joist to the same length, using a circular saw.
Hold the new joist, with an assistant's help, alongside the old one, flush against it.
Place the ends of the new joist into the same cutouts or braces that are holding up the damaged joist.
Hammer 3-inch nails through the new joist and into the old one. Sink nails every 18 inches or so along the top, bottom and middle of the joist. The new joist should completely brace and hold the weight of the floor while preventing any further movement by the damaged joist.