You can currently pick up a Nikon D600 Digital Camera with a 24-85mm f/3.5-4.5G ED VR Lens and other accessories for $1996. If you take the normal sale price of the lens ($597) and accessories ($174) away, you could say that the underlying price of the camera is $1225.
Getting very close to your $1,000 mark.
Another recent example is that B&H Photo is selling the Canon Rebel T4i w/18-55 IS kit for $649 . Minus the lens ($100 already highly discounted as a kit lens. Stand alone price is $199), that make the T4i body worth $549. Add $200 for the FF sensor, another $250 for larger body, mirror, shutter etc. That will make it $999 for a cheap FF body. Granted it will not be full of features. But it is doable.