I had the same problem, steve, and was trying to find some more detailed info to verify what I did. I got my info from a DIY book called Walls,Floors and Ceilings-- they rec. that you first install a new joist next to the rotted one. Locate it exactly 1 1/2" from the bad joist (the width of a 2x4). You'll need to chisel or route off the bottom 1/4" for 18" of each end, so you can get it over the sill plates or girders at either end. Once it's in place you hammer in some shims to make it fit tight against the subfloor, and toenail it in place. Once that's done you can rip out the old joist with a sawzall and crowbar. Cut all protruding nails flush with the sawzall. After all that you can replace the bad joist in its original location. same method, using shims to make it fit tight against the subfloor. Also you're supposed to put in some 2x4 blocks between the two joists you just put in (every 3 ft) to make it strong.
Finally, to resecure the floor to the joist go upstairs and nail in some 8d finishing nails. Of course, if your subfloor is also bad, it's an even bigger job.