Per Clarence House:

“Her Royal Highness The Duchess of Cambridge has been admitted to St. Mary’s Hospital, Paddington, London in the early stages of labour.

The Duchess travelled by car from Kensington Palace to the Lindo Wing at St. Mary’s Hospital with The Duke of Cambridge.”

According to BBC News journalist, Mr. Peter Hunt, the duchess was admitted to “…the private Lindo wing just before 6AM [London time].  According to Kate’s spokesperson ‘things are progressing as normal.'”