Actually, the body of the new Voldemort is his old body. No, it is not Tom Riddle's body, the young Tom Riddle we have seen in flashbacks, and from the diary - but yes it is Voldemort. Here are some supporting quotes:
"Voldemort had entered the room. His features were not those that Harry had seen emerge from the great stone cauldron almost two years before; they were not as snakelike, the eyes were not yet scarlet the face not yet masklike, and yet he was no longer handsom Tom Riddle. It was as though his features had been burned and blurred; they were waxy and oddly distorted, and the whites of the eyes now had a permanently bloody look, though the pupils were not yet the slits that Harry knew they would become." - HBP 413
IE. He is commiting more murders, and thus his soul is tearing, and he is becoming more and more transformed because of that. The more murders he commits, the more ugly he gets - until he becomes the snake-like Voldy we meet in GoF.
"You helped me return me to my body" - GoF 563
IE. It is not just any body, a brutal, misfigured body that he hasnt been in before. It is his, Lord Voldemort's body, that he has returned to; the body he has had since the last time he was powerful, killing enough people to mangle his soul.. making enough horcrxes to turn him into a monster.
Origionally posted by orionsixwings
The body that was animated in GOF is NOT Voldemort's original body. Therefore it, too, is a Horcrux. Remember that when the Avada Kedavra killed his body, his soul was already split in half. One went to --- my guess is ---Harry's scar, another floated elsewhere. It was this last piece that occupied Quirrell in Book 1. It is also this piece that occupied a fetus in Book 4. That fetus became a Horcrux. So the body that Voldemort is actually using now, is also a Horcrux.
When you cast Avada Kedavra, you don't send bits of soul flying around. There are a lot of people who have cast AK before, and they don't all have horcruxes. When you cast AK, you split your soul. You still need to do another spell, a spell we don't know, to encase it in an item that will conceal it - a horcrux, as Slughorn tells us:
"A horcrux is the word used for an object in which a person has concealed part of their soul. You need to split your soul to make a horcrux. How? The supreme act of evil. By committing murder. Killing rips the soul apart. The wizard intent on creating a horcrux would use the damage to his advantage: he would encase the torn portion in an item. How? There is a spell, but do not ask me, i don't know it." - Proffessor Slughorn, HBP, chapter: Horcruxes.
This is proof that you need to do a horcrux spell, after committing the murder. If we are to believe that, as Dumbledore said, Voldemort was going to use the very significant death of Harry's as the final tear, the final horcrux to complete his 7, then he would have had to cast the horcrux spell afterwards, to conceal his soul within an item (or possibly Harry). He did not do that, and thus, Harry cannot be a horcrux.
On the other hand, if you are arguing for Harry being a horcrux, Nagini being one is very strong evidence for this. Voldemort can posess Nagini, like we have seen him posess Harry - and all the while, whilst Voldemort is in his actuall body, and not a spirit anymore. It is possible that possession, by a wizard who has a body, is very very unusual, maybe impossible in the wizarding world - which would lead to Dumbledore's famous question "but in essence divided?"
He says this right after the snake attack on Mr Weasley. This, i believe, is proof that Nagini, in this instance, was being possessed by Voldemort. Further proof of this is the fact that Harry saw the dream through the snakes eyes. He can't get into snakes minds like this - but he can get into Voldemorts. Indeed, we have seen him dream of being Voldemort before, seeing from Voldemorts perspective, like he saw from the snakes perspective in OoTP.
If we take it that when Nagini attacked Arthur, it was actually Voldemort, then Voldemort can possess both Nagini and Harry. Because they both contain a portion of his soul? Possibly. It is also possible that Voldemort can simply possess snakes, but i don't know. I think the silver instrument that Dumbledore questioned is a very key scene. It showed a snake, and when he asked the question, it divided into two: Nagini and Voldemort. He concluded that Voldemort possessed the snake, i think. This may also be where he gets evidence to his claim that Nagini is a horcrux.