No it wouldn't fit nicely into an Exige. And yes I am familiar with them and making things fit, see my avatar, thats mine and 450hp. An exige comes with a sideways mounted 1.8L. The Veyron is a North/South 16 Cylinder with 4 turbos. You would have to cut the Exige right in half and add another half a car to it's mid section, as the engine is too long to jam is sideways, it would have to be north south, and when you do that you need to have a tranny that also adds to the length.
Think of the Exiges setup as a Front wheel drive car, it's basically a Celica, except flipped backwards so the motors in the back. Think of the Veyron as a VW Phantom except motors i nthe mid, and it's in the mid because its so large with a tranny sticking out the back. Completely different layout.
Anythings possible with a car, of course, but I wouldn't call it an Exige if it doubled in length, as then there is no point, the handling would be gone, the weight would be massive, the looks would be butchered.
Nothing is kidding in the world of automobiles as anything can be done, all I was saying is, it would be pointless as it would wreck the Exige. Where as you had made it sound like it would retain its character while gaining 980hp, when it would in reality be an entirely different car.
http://img.techpowerup.org/101207/Capture012.jpg
Back engine is an Exige S motor, other is the Veyrons, they aren't exact scale, but thats pretty darn close, The Veyron has 30inches more distance between the wheels, thats 2.5ft of extra engine space. Which the door is maybe 30in long, and that motor looks to just about match up with it, which would seem correct.