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There are 9/4 odds that the royal baby will be born in the third week of July, bookmakers William Hill said.
If the new prince or princess were born on July 1, he or she would share a birthday with William's mother, the late Diana, Princess of Wales.
The Duke and Duchess of Cambridge are unlikely to undertake any engagements in the next few weeks as William will be concentrating on his flying with the RAF Search and Rescue Force, a spokesman for the couple said.
He added that Kate is likely to return to engagements "in the near future".
The Duchess's condition is continuing to improve following her earlier hospital stay for severe morning sickness, a St James's Palace spokesman said.
The royal baby is due in July, St James's Palace announced.
It was announced that the Duke and Duchess of Cambridge were expecting their first child in early December.
The announcement was made less than 12 weeks into the pregnancy after Kate was admitted to the King Edward VII Hospital in central London suffering with Hyperemesis Gravidarum - a severe form of morning sickness.
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As it is confirmed the Duke and Duchess of Cambridge's child is due in July, researchers look at what the future holds for the summer baby.
The Duke and Duchess of Cambridge's baby is due in July, St James's Palace confirmed.