A FlyerTalk Posting Legend
Join Date: Jul 2002
Location: MCI
Programs: AA Gold 1MM, AS MVP, UA Silver, WN A-List, Marriott LT Titanium, HH Diamond
Posts: 53,012
I did exactly that last year.
It's a trade-off: they'll do it entirely in one call with no credit pull, but you don't get any sign-up bonus on the Prestige card. If you want the sign-up bonus, do a new application and credit pull...and then cancel the Exec card.