Spencer, 37, was part of a 17-strong team commissioned by National Geographic to survey and film both outside and inside the 120m-deep wreck, from 19-31 May.
He was on the wreck when he got into difficulty and, despite efforts made to help him, was dead by the time he reached the surface.
Diving operations have been suspended and the project team questioned by the Greek authorities. Spencer's equipment, which includes a rebreather, is being held by the authorities, and it is not yet clear where and by whom it will be officially examined.
Spencer was a veteran of three expeditions to the Britannic. One which he led in 2003 provided evidence that the ship had been sunk by a mine rather than a torpedo, a question that had vexed historians.
A sister ship to the Titanic, the Britannic was launched in 1914 with design modifications introduced after the Titanic’s sinking two years earlier. The ship never operated as a liner, outfitting being delayed before requisitioning in 1915 as a hospital ship.
The ship was sunk in November 1916 four miles off the island of Kea, south-east of Athens, while en route to evacuate soldiers from Mudros, on the Greek island of Lemnos. Of 1066 people aboard, 30 lost their lives when two lifeboats were drawn into its propellers.
The wreck, which lies on its starboard side and is well preserved, is owned by British wreck historian Simon Mills. He has planned to run tourist dives to the vessel in mini-submarines.
In 2003 Spencer also took part in deep Atlantic submarine dives to the wreck of the Titanic, as part of a Discovery Channel filming expedition led by director James Cameron.
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