The Duchess is entitled to use the title Princess William of Wales, but has never described herself as such because the couple decided to be known as the Duke and Duchess of Cambridge after the titles were conferred on them by the Queen on their wedding day.
The couple were given a standard birth certificate for "His Royal Highness Prince George Alexander Louis of Cambridge", but unlike other new parents they did not have to attend their local register office. Click link for full post from The Telegraph