Described as the most authentic replica ship in the world, it took six years to construct from original plans and papers stored at the National Maritime Museum in Greenwich, south-east London, and was funded by the Endeavour Foundation in Fremantle, Western Australia.
The ship left Australia for its grand voyage to Britain last October and will now berth at Greenwich for two weeks before embarking on a coastal tour including stops at Great Yarmouth, Whitby, Inverness, Liverpool, Fishguard and Plymouth.
The first Endeavour was built in 1765 at Whitby, North Yorkshire, where Cook trained in maritime navigation. Although obliged by law to have radar and back-up engines, the modern counterpart is sailed in much the same way as when Cook left Britain in 1768. There are laminated masts and synthetic canvas, but below decks the modern Endeavour is arranged exactly as it was in his time.
This was the ship in which he set sail to observe the transit of Venus across the sun over Tahiti. In New Zealand, he and his crew became the first Europeans to record Maoris. Joining the very select band of Europeans to have visited Australia, in 1770 they discovered the east coast - setting the foundations for the colonisation of Australasia and seeing for the first time Aborigines, koalas and kangaroos.
But life on board was tough. Of the 94 men who sailed, only 41 survived the voyage and the ship came close to being wrecked as the crew struggled to navigate the Great Barrier Reef.
The Queen and the Duke of Edinburgh are due to visit the replica ship today. They will meet some of the 52 crew from Britain, Australia, New Zealand and South Africa, including some who have been on board for the full 115 days at a personal cost of pounds 4,500.
Places are still available for manning the ship as she sails between British ports. Applicants must be over 18, have some experience of the sea and a good head for heights.