George V’s declaration sets out: “The grandchildren of the sons of any such sovereign in the direct male line (save only the eldest living son of the eldest son of the Prince of Wales) shall have and enjoy in all occasions the style and title enjoyed by the children of dukes of this realm.”
As such, Archie will be entitled to the titles when Prince Charles accedes the throne.
George V’s declaration means that only Prince George, as a great-grandson of the monarch down the direct line of succession to the throne, was originally entitled to be a prince, as he is the eldest son of the eldest son of the Prince of Wales.
The Queen did step in ahead of George’s birth to issue letters patent to ensure the Duke and Duchess of Cambridge’s children would have the titles of prince and princess.