Prince William arrives at King Edward Hospital on Dec. 4
12/04/2012 at 01:45 PM EST
returned to the hospital
to visit pregnant wife Kate
as she spends her second day being treated for hyperemesis gravidarum, a severe type of morning sickness
William – in a purple sweater, blue collared shirt and black jeans – walked straight into King Edward VII Hospital after exiting a green Land Rover.
Without a wave or signal to the 100-strong waiting media, the father-to-be
, 30, had his head down, briefly running his hand through his hair as he strolled through the doors.
He stayed at the hospital for about six hours (until just before 6 p.m.), and left with a smile but did not address the press who had gathered outside.
Kate, who is believed to be being cared for by Dr. Marcus Setchell (he delivered royal babies for the Countess of Wessex, William's aunt), was brought to the hospital by William on Monday after her sickness became more acute.
Meanwhile, Palace sources tell PEOPLE that Queen Elizabeth
, who was only told Kate was expecting once the couple had decided to head to hospital, will be unlikely to visit.
"She doesn't normally go because it disrupts the running of the hospital," the source said. "And the Duchess just wants to recover and get out."
Kate's office has canceled all her planned engagements this week. She was due to appear at a charity day at a London bank Wednesday, a gala for the Centrepoint homeless charity and visit the military tournament this weekend.
On Tuesday, a St. James's Palace spokesman announced her condition is improving.
"The Duchess of Cambridge is continuing to feel better," he said. "She and the Duke are immensely grateful for the good wishes they have received. She will remain in hospital at present and will continue to be treated for hyperemesis gravidarum."
This will be the first child for the couple, who wed in April 2011