It is very simple and fast to transfer private registration from one car to another. You will need to fill out a V317 form in order to transfer a private number plate. This can be picked up at your local DVLA office, or at your main Post Office. 

The details of both the donor vehicle (the vehicle that currently carries the registration number) and the recipient vehicle will be required. Vehicles must have valid MOT certificates and current licenses in order to transfer vehicles.

If you want to register your car then you can look for vehicle registration online services.

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You will need to attach the V5/V5C logbooks for both vehicles involved in the transfer, MOT certificate for each vehicle if applicable, and photocopies of each tax disc. The transfer fee is currently at PS80.

Transfer applications may be made for unlicensed vehicles provided that the license was not expired more than 12 months ago and that the vehicle has been declared SORN. There is no gap between the SORN declaration's commencement and expiry date. 

Don't panic if you have no other vehicle to give your personal registration to, but need to remove your vehicle's personal registration. Private number plate owners can keep their certificates on the DVLA retention program, which allows them to be assigned to a vehicle without having to take their personal registration off of their vehicle.