Who will be the next Duke of Edinburgh?

It was announced around the time of Edward's wedding that he would be granted the title after his father's death.

I don't have a solid theory as to when the Queen will officially do that, but I would guess that it wouldn't be any earlier than after her husband's funeral.

It was automatically inherited by Charles.  The agreement made in 1999 wasn’t that Edward would inherit it, but that he would get the title IN DUE COURSE. Meaning that when Charles becomes king, provided he sticks to the agreement, he will issue new Letters Patent creating the title for his youngest brother.  

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Prince Edward, the Queen's youngest son.   Charles & Andrew already have Duke titles and Prince Edward just has Earl of Wessex.

It most certainly won't be a troll like you.

It has already been said that Prince Edward would take the title of Duke of Edinburgh on the Duke's death.   Most probably decided a while ago by HM.   This is probably why Edward wasn't given a Dukedom on his marriage, as others of the Family were.

Edward will NOT become DofE until AFTER the Queen passes. The title will be bestowed upon him by Charles, who will by that time be King.

At the time of Prince Edward's wedding to Sophie Rhys-Jones, it was announced that Edward would one day be bestowed Duke of Edinburgh. The eldest son of a duke usually inherits the title, but in this case, Charles, Prince of Wales, Duke of Cornwall, Duke of Rothesay, future monarch, agreed that his younger brother should be Duke of Edinburgh; after all, Edward has been working with his father for decades on the Duke of Edinburgh schemes and wildlife conservation efforts.

Charles will let the title go into abeyance so his brother can be bestowed the title. This should not take a long time--probably after a period of mourning, the Queen will bestow Edward Duke of Edinburgh.

It has long been thought, that the title would be recycled and used by Prince Edward.

It was decided a while again that Prince Edward the youngest son would have the title

The next one will be Donald Trump.