Selling car and transferring private registration in one go?

[oxtailsoup]

Selling my car which has my private registration on.

Can I complete the V317 with my details to transfer the private plate to my new car, and on the same form fill out the new keepers details for the existing car. All in one go?

And send the V317 and the V5C to DVLA....

I seem to be finding conflicting info on the web but it appears from the V317 form that this can be done in one go rather than transferring the plate weeks before selling it.