According to the BBC, a statement from Clarence House said that the Queen and members of both families are "delighted with the news".
David Cameron and Ed Miliband offered their congratulations.
Sky News reported that the duchess is currently being observed by doctors at Kensington Palace due to her acute morning sickness, and that the duke will continue a planned engagement in Oxford today alone.
AP reported that the new baby will become fourth in line to the throne. Prince Harry will now be fifth.
According to the Daily Mail:
William, 32, recently announced his decision to put off becoming a full time working royal in favour of taking a new role as an East Anglian air ambulance pilot.
The new role will allow the couple to bring up their family in the relative privacy of their new home in Norfolk, Anmer Hall.
The betting on what their second baby will be called has been going on since the first was born.
Paddy Power is offering 10/1 on the child being named Elizabeth, Henry, or Victoria – staunch and traditional royal names – but anyone fancying a flutter could also try Charles at 12/1, James at 16-1, or the 20/1 shot Diana.
Way out at the back of the betting pack is the extremely unlikely Camilla, at 100-1.
The Duke of Cambridge was seen opening the Dickson Poon University of Oxford China Centre building.