RW, if you can use the same supports that the current joist uses, the job should be relatively easy. Get some pressure treated wood the same size as the current joist and sister one in on each side. Be sure to go all the way across from one pier or sill plate to the other. You want to have the sisters have as much support from underneath as the current joist. Us a jack of about any kind ( even a car's bumber jack will probably work) to take some of the load off the current joist. You don't want to lift it, just take the load off when you nail the sister into the old joist. This sounds like a lot of work but should be really easy.
A bigger problem might be finding out where the water is coming from that caused the rot. Recaulk around the tub and check for cracks in tile grout and leaky pipes and valves. Hope you are lucky because the water might have caused a lot more problems that just the rotted joist.