Like GolfAndWoodNut and Beetle-Kill suggested, installing two joist headers and hangers to support the cut ends of the modified joist(s) is the way to go.
I'm not sure how clear this will be, but:
Remove the drywall ceiling below for about 4 to 5 feet around the spot to be cut, nail 2x4 plates perpendicular to the joists on each side of the section to be cut out and spanning both the joist you want to cut and the neighboring joists on each side. Then put bottom plates directly below them on the floor, and hammer in some 2x4's cut about 1/8 inch taller than the space between the top and bottom plates. That will result in two 'pressure fit' supporting walls oriented perpendicular to the joists, one on each side of the soon to be cut area of the target joist. You'll want one stud 2x4 below each joist. Cut out the section out of the middle joist, install the headers, through nail it all, and then slip on and nail in the joist hangers (one at each end of each header and one at each point where the cut joist meets a header, for a total of 6). Take down the supporting 2x4s, pull the top and bottom plates, patch the drywall, install your through-floor fitting, and you're golden!
Since you're only supporting a single joist, you could probably get away without sistering the joists that the headers hang from, though it would be safest to have a doubled joist on each side running all the way from one joist support point to the other. That would obviously require some additional drywall removal and should be done before you take any of the steps listed above after cutting out the drywall.
If you don't have dimensional lumber joists and instead have engineered wooden joists that look like I-beams, make sure you install a filler piece to bring the web flush with the beams in each place that a header or joist butts up at 90 degrees to the engineered joist. The joist hanger needs solid contact to do it's job and bear the weight of the hanging joist.
And, my disclaimer: If any of that doesn't make sense, call a contractor!
