The Duchess of Cambridge faces a third day in hospital as she is treated for a severe form of morning sickness.
Kate was "continuing to feel better", St James's Palace said on Tuesday night.
Relief of his pregnant wife's improving condition was evident in William's smile as he emerged from the hospital after spending the day at her bedside.
The Duchess was admitted to London's Edward VII hospital on Monday, just before news of her pregnancy was revealed. She has developed the condition hyperemesis gravidarum, which can cause severe vomiting and leave patients feeling tired, dizzy and suffering from headaches, caused by dehydration.
But it now appears she is responding to treatment and is likely to be having less severe bouts of morning sickness. It is likely she will be on a drip so she can receive fluids intravenously to combat the effects of dehydration.
The Queen's former gynaecologist Marcus Setchell, who delivered the Countess of Wessex's two children, is treating her, according to reports.
Kate's illness could indicate she is having twins as mothers carrying two babies have a greater chance of developing the severe morning sickness.
William spent more than six hours with his wife in hospital on Tuesday and left at 5.45pm looking relaxed, smiling at the waiting reporters and photographers.
Kate's diary of engagements for this week have been cancelled to allow for the extensive rest she will need when she is discharged.
The prolonged vomiting could continue for much of her pregnancy - which is still below 12 weeks - and she may need to take anti-sickness medication for months to come.