New princess to meet Her Majesty, Great-Grandma

The Little Princess showed up for her first official royal engagement -- fast asleep -- just a few hours after she was born on Saturday morning.

The royal baby, her name -- if she has one yet -- still a mystery to the world, is fourth in line to the British throne, and a second child for proud parents the Duke and Duchess of Cambridge.

CBS News correspondent Elizabeth Palmer reports Queen Elizabeth herself, now 89, hasn't met her great granddaughter yet, but she's about to.

The Queen was at Sandringham on Monday, her rural estate in Norfolk, England. The Duke and Duchess of Cambridge were thought to be headed that way later in the day, where they'll likely stay at their own secluded country retreat nearby, Anmer Hall.

The brand new princess has, however, already met her big brother Prince George, who arrived at the hospital for a visit on Saturday with William. The family left St. Mary's Hospital the same day.

Later, other members of the royal family came to the couple's official residence in London, Kensington Palace, to welcome the first ever British princess born with equal rights to wear the crown.

"The law has changed in this country now to give princesses, if you like, a fairer crack at the whip, and they can't be superceded by princes," Royal biographer Robert Hardman told CBS News.

The name game has kept Brits guessing all weekend -- and Britain's bookies doing a brisk business. The smart money is betting on tradition.

The name of Prince William's late mother Lady Diana is in the mix, says Palmer, but it's considered a long shot thanks to her public falling out with the royal family and her sudden death.

As Hardman says, they need to choose a name that will stand the test of time.

"Prime ministers and presidents are known by their surnames, but the royals are known by their Christian names. This name is going to resonate," the author told CBS News. "It will be a part of British history for a century."

Palmer says the now-enlarged royal family is expected to lie low for the next week or so in Norfolk.

/By CBS News/