Led by Mr Hyett, the group dredged the bars from the ocean floor before putting them in storage in a Southampton warehouse believing they were commercial property rather than government-owned.
However, the South African government subsequently claimed ownership of the silver, triggering a lengthy legal battle over who had rights to the silver and the cost of the expedition.
While Argentum eventually admitted the precious metal was the property of South Africa, it had been seeking to recoup the cost of recovering the cargo from the bottom of the sea.
However, in a landmark ruling on Wednesday, the UK Supreme Court found South Africa was immune from the claim.
Argentum and South Africa reached a confidential settlement shortly before the judgement was handed down.
A spokesman for Argentum said: “The Supreme Court has overturned the decision of both the High Court and Court of Appeal in an important judgement clarifying the law on sovereign immunity and its applicability in an in rem salvage claim.
“The parties concerned have now reached a settlement which brings the matter to a close.”
Mr Hyett, an entrepreneur and property magnate, founded Argentum with the special mission of recovering the SS Tilawa’s silver. Sir Paul is a shareholder in Argentum but was not involved in the operation.
Sir Paul co-founded hedge fund Marshall Wace and co-owns TV channel GB News.
He is among the bidders interested in The Telegraph, which is up for sale after UAE-backed media group RedBird IMI was blocked from owning the title as a result of its links to the Abu Dhabi government.