So this is adjacent to the foundation wall? If so, you can certainly bear the header on a piece of material bearing on the foundation wall, or sneak it up on the mudsill if you can.
Cut out the rotten part and use pieces of lumber of the same dimension (2x10 or whatever the joist is) perpindicular to the joists. I'd suggest using joist hangers on the header pieces' connections to the joist, and also at the connection of the tailjoist to the header.
Just remember that the 3/4" floor sheathing cannot span more than 24" in any direction...So a 24x24" area would be the biggest area you can have that is not supported by a joist. Hope that makes sense.