Carole and Michael Middleton were the first family visitors at the hospital since the birthThe couple earlier thanked hospital staff for their "tremendous" care.
It is understood the family will leave hospital on Tuesday evening or on Wednesday morning.
Gun salutes in Green Park and the Tower of London and the ringing of bells at Westminster Abbey have 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, rung for three hours.
The Middletons arrived at private Lindo Wing of St Mary's Hospital, in Paddington, west London, by taxi shortly after 15:00 BST.
They were greeted by its lead clinician, Jonathan Ramsay, and paused briefly on the steps for the mass of photographers gathered outside before going inside.
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.
There has been no word yet on what the couple plan to name the prince.
The Duchess of Cambridge remains in hospital following the birth of her son on Monday. Reporters and photographers are camped outside the Lindo Wing of St Mary's in Paddington, west London
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
Prince Charles was cheered as he undertook duties in Bridlington, East Yorkshire. There were cries of "Congratulations, grandad" from the crowd
Even the ship's company of the Royal Navy frigate HMS Lancaster found time to mark the occasion while on patrol in the CaribbeanAt Buckingham Palace, royal watchers and tourists are queuing 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.
Fresh from trying to get her photo of the easel, Robyn Holtham, 41, from Oxfordshire, described the birth as fantastic news.
"The Royal Family has been becoming more accessible with Kate and William, people can identify with them. Kate is a more normal person. They're just people who like to have fun and do normal things," she said.
'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.
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:
- New Yorkers were informed of the news when it was tickered in lights at Times Square
- In London, Trafalgar Square was lit blue for a boy, while the BT Tower delivered the message: "It's a boy!"
- 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
William and Catherine - said by officials to be "so, so happy" - spent the night, a thundery and wet one, with their newborn son in their private maternity suite.
After a labour which had lasted at least 10 hours - and despite the clamour for information from an expectant world - the couple did what most new parents do. They spent time alone bonding with their son.
Soon, they will have to leave their cocoon and present their prince to those eager for a glimpse of the baby.
When they leave, they will be photographed on the steps of the Lindo Wing, just as baby William was back in 1982.
His son's birth will be marked with gun salutes and the pealing of church bells.
It is a celebratory reception the one-day-old royal will have to get used to.
He is a prince who one day, as things stand, will be a king.
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.
Nicholas Witchell said the third in line to the throne could expect to be brought up in a "secure and loving environment", shielded from many of the pressures of ordinary life but facing the specific challenges his position will bring.
The baby's future, he said, will be "a lifetime of public curiosity and, in due course, the responsibility of refreshing and taking forward the ancient institution that is the world's best-known hereditary monarchy".
bbc.com