Royal baby: Kate and William visited by Prince Charles

Prince Charles has visited the Duke and Duchess of Cambridge and his new grandson in hospital in west London.

The arrival of the Prince of Wales, accompanied by the Duchess of Cornwall, came two hours after the duchess's parents, Carole and Michael Middleton, became the first visitors.

The prince said the baby was "marvellous" while Mrs Middleton said he was "absolutely beautiful".

The new family are to leave hospital this evening for Kensington Palace.

Speaking to reporters as she left the hospital, Mrs Middleton said mother and baby were "both doing really well - we are so thrilled".

She added her first cuddle with her new grandson had been "amazing" and said the new parents were doing "fabulously".

There has been no word yet on what the couple plan to name the prince and asked if she had been told or made any suggestions, Mrs Middleton said: "Absolutely not".

Earlier, gun salutes in Green Park and the Tower of London and the ringing of bells at Westminster Abbey marked the birth of the third in line to the throne.

The King's Troop Royal Horse Artillery fired a 41-gun salute in Green Park at 14:00 BST, after riding past Buckingham Palace.

At the same time, the Honourable Artillery Company - the City of London's army reserve regiment - fired a 62-gun salute from Gun Wharf at the Tower of London.

Meanwhile, the church bells of Westminster Abbey, where William and Catherine were married in April 2011, rang for three hours.

Hundreds of journalists and members of the public are gathered outside the private Lindo Wing of St Mary's Hospital, in Paddington, where the Middletons arrived by taxi shortly after 15:00 BST.

Prince William was at the hospital for the birth at 16:24 BST on Monday, and stayed with his wife and son overnight.

Kensington Palace said in a statement: "Mother, son and father are all doing well."

The couple said: "We would like to thank the staff at the Lindo Wing and the whole hospital for the tremendous care the three of us have received.

"We know it has been a very busy period for the hospital and we would like to thank everyone - staff, patients and visitors - for their understanding during this time."

Prince William had earlier said the couple "could not be happier" following the birth of the 8lb 6oz boy.

The Duchess of Cambridge remains in the Lindo Wing of St Mary's Hospital after the birth of her son. Reporters and photographers are camped outside

Prince Charles and the Duchess of Cornwall arrived at the hospital after completing engagements in Yorkshire

The duchess's parents, Carole and Michael Middleton, were the first relatives to visit the couple - and their first grandchild

Well-wishers have been queuing up outside Buckingham Palace to take pictures of the official birth announcement

The Honourable Artillery Company fired a 62-gun salute at the Tower of London in honour of the birth

The King's Troop Royal Horse Artillery fired a 41-gun salute in Green Park and bells were rung at Westminster Abbey

Even the ship's company of the Royal Navy frigate HMS Lancaster found time to mark the occasion while on patrol in the Caribbean

At Buckingham Palace, royal watchers and tourists have queued in the rain to catch a glimpse of the bulletin announcing the arrival of the royal baby, which is being displayed on an easel.

The news broke about four hours after the birth and the notice is to be displayed for 24 hours, which means it is likely to be removed just after 20:00 BST.

Bandsmen of the Scots Guards outside the palace brought a celebratory feel to the proceedings ahead of the Changing of the Guard, playing a version of Cliff Richard's Eurovision song Congratulations to the crowd.

'Knocked back'

The world now awaits the couple's choice of names for their son, with George the bookmakers' favourite, followed by James and Alexander.

BBC royal correspondent Nicholas Witchell said the scale of the international interest was "pretty awesome", and "quite threatening in a sense". He said the couple would be "knocked back" when they stepped out of the hospital.

The arrival generated headlines and celebrations around the world, and prompted messages of goodwill to flood in:

• In London, Trafalgar Square was lit blue for a boy, while the BT Tower delivered the message: "It's a boy!"

• New Yorkers were informed of the news when it was tickered in lights at Times Square

• At its peak, there were more than 25,000 tweets per minute about the royal baby's birth

• US president Barack Obama said: "We wish them all the happiness and blessings parenthood brings" while Australian PM Kevin Rudd said Australians wished "the royal bub all the best"

• The Archbishop of Canterbury, Most Reverend Justin Welby, greeted the news by saying: "Along with millions here and around the world, I share in their joy at this special time"

• A specially-filmed scene will be inserted into Tuesday's edition of EastEnders to mark the occasion

'Uplifting moment'

Following the birth announcement, a statement from Kensington Palace said the Royal Family were "delighted".

The Duchess of Cornwall, on a two-day visit to Yorkshire with the Prince of Wales, said it was a "wonderfully uplifting moment for the country" and that mother and baby were "doing well".

The prince said he was "thrilled and very excited", as the couple were congratulated by well-wishers during a walkabout.

Meanwhile, Princess Anne said the arrival of the Royal Baby was "nothing to do with me, but it's very good news".

Royal doctor Mr Setchell described the new arrival as a "wonderful baby, beautiful baby".

The Leader of the House of Lords, Lord Hill of Oareford, said the birth represented a "very happy day" for the whole country.

"It is the crowds who have once again gathered outside Buckingham Palace which are the real mark of the great affection in which the Royal Family is held," he said.

The birth of the prince means the monarchy has three generations of heirs to the throne for the first time since 1894.