It depends, if the chip has not been soldered, but placed in an IC (DIP) socket, than you can remove the programmed chip, and use it in another circuit and place another (same) IC in the IC socket of the Uno to be reprogrammed.
You also could desolder it, but I would not advice that; also most of the Uno's having a soldered IC use SMD instead of DIP.
Also, to replace the IC, you need to install a bootloader on the IC to be placed in the new Arduino (thanks for the addition Roberto Lo Giacco).
Below is a version with DIP that easily can be replaced, hence the big IC and the socket:
Below is a version with SMD that cannot easily be replaced, hence the small IC without a socket: