Cook famously sailed the ship around the South Pacific before landing on the east coast of Australia in 1770.
Australian National Maritime Museum CEO Kevin Sumption announced that after a 22-year program of archival and archaeological research, "we can conclusively confirm that this is indeed the wreck of Cook's Endeavour".
"This is an important moment," he told reporters at National Maritime Museum in Sydney on Thursday.
"It is arguably one of the most important vessels in our maritime history."
The ship played an important role in exploration, astronomy and science and was an important artefact in the history of Australia, New Zealand, the United Kingdom and now the US, he said.
A "preponderance of evidence" had led to the conclusion that an archaeological site known as RI2394 in Newport Harbour, Rhode Island, "does indeed comprise of the shipwreck of HM Bark Endeavour", he said.
But not everyone agrees - yet.
Rhode Island Marine Archaeology Project executive director Kathy Abbass says the Maritime Museum has jumped the gun in declaring the wreck the Endeavour because RIMAP "is now and always has been the lead organisation for the study".
Dr Abbass says the announcement breaches a contract between the two organisations, and despite finds "consistent with what might be expected of the Endeavour" there has been "no indisputable data" yet found to prove it "and there are many unanswered questions that could overturn such an identification".
A "legitimate report" will be posted on the RIMAP website when the study is complete, Dr Abbass says.
"RIMAP recognises the connection between Australian citizens of British descent and the Endeavour, but RIMAP's conclusions will be driven by proper scientific process and not Australian emotions or politics," she added.
A spokesperson for the National Maritime Museum says it has not breached any contracts and "we look forward to pursuing a due process of peer review and consultation with all stakeholders in Rhode Island".
"Having worked with Dr Abbass to investigate the Endeavour shipwreck site for 22 years, the museum acknowledges that she is entitled to her own opinion regarding the vast amount of evidence we have accumulated," the spokesperson said.
Since 1999, maritime archaeologists had been investigating several 18th century shipwrecks in a five square kilometre area of Newport Harbor, where the ship was scuttled 244 years ago.
Although only about 15 per cent of the vessel remains, several details on the wreck convinced archaeologists they had found Endeavour after matching structural details with 18th century plans of the ship.
Researchers are now focused on what can be done to protect and preserve the historic remains.
Mr Sumption said the museum was looking to "borrow some material at a future date to come to Australia and be on display".
Originally launched in 1764 as the Earl of Pembroke, the ship was renamed Endeavour in 1768 by Britain's Royal Navy and prepared for a major scientific voyage to the Pacific.
From 1768 to 1771 Endeavour sailed the South Pacific, primarily to record the transit of Venus in Tahiti in 1769.
Cook, who was also an acclaimed navigator and cartographer, then sailed it around the South Pacific searching for "the Great Southern Land", charting the coast of New Zealand and Australia's eastern coastline before claiming the land for Great Britain on August 22,1770.
