Video report by ITV News Royal Editor Chris Ship
The Prince of Wales thanked the nation and people across the Commonwealth for the outpouring of loss and "touching things which have been said" about Prince Philip, which he said would "sustain" the Royal Family "at this particularly sad time".
Watch Prince Charles' tribute to his father
Speaking from outside his home of Highgrove in Gloucestershire on Saturday, Prince Charles said he and his family "miss my father enormously" as he paid tribute to a "very special person".
"I particularly wanted to say that my father... I suppose the last 70 years has given the most remarkable, devoted service to the Queen, to my family and to the country and also to the whole of the Commonwealth and as you can imagine my family and I miss my father enormously," the Duke's eldest son said.
ITV News Royal Editor Chris Ship analyses Prince Charles' tribute and looks ahead to the funeral
"He was a much-loved and appreciated figure and apart from anything else I can imagine that he would be so deeply touched by the number of other people here and elsewhere around the world and in the Commonwealth who also think... share our loss and our sorrow and my dear Papa was a very special person, who I think above all else would have been amazed by the reaction and the touching things that have been said about him.
"And from that point of view, we are, my family, deeply grateful for all that.
"It will sustain us in this particular loss and at this particularly sad time. Thank you."
Watch: The remarkable life of the longest-serving royal consort in British history
As reflected in Prince Charles' statement, the Duke was the longest-serving consort in British history and dedicated decades of his life to royal duty, serving the nation at the monarch’s side.
Announcing his death in a statement on Friday, Buckingham Palace said: "It is with deep sorrow that Her Majesty the Queen announces the death of her beloved husband, His Royal Highness the Prince Philip, Duke of Edinburgh.
"His Royal Highness passed away peacefully this morning at Windsor Castle.
"Further announcements will be made in due course. The Royal Family join with people around the world in mourning his loss."
The tribute from Prince Charles came shortly after funeral plans for his father were announced.
The date has been confirmed as Saturday, April 17 and Covid restrictions mean the event will be scaled-back and will not feature a public procession.
The ceremony will take place at Windsor Castle and will be broadcast live, including on ITV at 3pm.
There will be no lying in state and no state funeral for Prince Philip, according to his personal wishes which were taken into consideration and approved by the Queen.
ITV News Royal Editor Chris Ship has more on the funeral plans
His coffin will be transported to St George's Chapel at Windsor Castle in a Land Rover designed with the help of the Duke of Edinburgh.
Only the Royal Family and the Duke’s private secretary will attend the ceremony inside the chapel, leaving others within the grounds to disperse in silence.
The Duchess of Sussex is heavily pregnant and has been advised by her doctors not to travel to the UK for the funeral, it is understood.
Currently, Covid rules in England restricts funerals to 30 guests, which will be adhered to by the Royal Family, although military personnel will be excluded from this figure as they will be deemed to be on duty.
The coffin will be draped with the Duke of Edinburgh’s insignia personal standard, in addition to his naval cap, sword and a wreath of fresh flowers.
Although the event will be a mournful once, the service will celebrate the Duke's life and achievements in his 99 years, the Palace said.
At the conclusion of the service, the coffin will be entered into the Royal Vault, where it will remain until the death of the Queen, whereupon they will be buried together.
The general public have been told not to travel to Windsor.
A Palace spokesperson said the Royal family hoped the coming days would be seen as a chance to celebrate the duke’s “remarkable life”.
Flags at royal residences will continue to fly at half-mast until the morning after the Duke is laid to rest.
The Royal Family will be observing two weeks of formal mourning, that began on Friday, the day of his death.
Earlier on Saturday, gun salutes took place across the UK, in Gibraltar and at sea to mark Prince Philip's passing.
Watch the gun salutes taking place across the country
Saluting batteries fired 41 rounds at one round every minute starting at midday in cities including London, Edinburgh, Cardiff and Belfast, as well as Gibraltar and from Royal Navy warships, the Ministry of Defence (MoD) said.
Gun salutes have been fired to mark significant national events since as early as at least the 18th century.
Prince Philip's son, the Earl of Wessex, was seen arriving at Windsor Castle following the death of his father on Saturday morning and left with his wife Sophie around an hour later.
As she left Windsor Castle, the Countess of Wessex said “the Queen has been amazing”. Sophie spoke to reporters from a Land Rover driven by her husband.
Watch Prince Edward and his wife Sophie leaving Windsor
Edward's brother, Andrew, was also seen leaving Windsor shortly after.
It is understood Prince Charles travelled to visit the Queen on Friday afternoon, travelling from his home in Gloucestershire to be by his mother's side at Windsor Castle.
As well as the statement made by Prince Charles, other members of the Royal Family have been sharing their memories and paying tribute to the Duke.
The Royal Family's Instagram account shared a moving quote made by the Queen during the celebration of their golden wedding anniversary in 1997.
She said: "He (Philip) has, quite simply, been my strength and stay all these years, and I, and his whole family, and this and many other countries, owe him a debt greater than he would ever claim, or we shall ever know.”
In an ITV documentary broadcast on Friday night, the Duke of Edinburgh’s children paid tribute to him, describing him as the rock in the Queen and their family’s lives.
The Princess Royal and the Earl of Wessex reflected on their loving relationship with their father and the nation's longest serving consort.
Paying tribute to His Royal Highness, they shared their personal childhood memories of the Duke and the Queen, the famed Duke of Edinburgh Award and their father's lasting legacy.
Princess Anne speaks to ITV News Royal Editor Chris Ship about her father, the Duke of Edinburgh
Prince Harry and Meghan paid tribute to Philip on the website of their foundation Archewell, replacing its homepage with a memorial site and the words: "In loving memory of His Royal Highness The Duke of Edinburgh 1921-2021.
"Thank you for your service…You will be greatly missed."
The Royal Family has asked members of the public consider making a donation to a charity instead of leaving floral tributes in memory of the Duke of Edinburgh.
An online Book of Condolence for those who wish to leave messages is available on the Royal website.