From the notes of the album 'Leviathan!':

[1967:] Chase of the right whale, West Greenland, 1820s. As a result of merciless hunting, the greater number of whales had migrated westward from Spitsbergen by the end of the 18th century, and were found off the east coast of Greenland. But by the 1820s, they began to move westward again, into the Davis Straits. [By 1830,] Peterhead owned thirteen whalers, Aberdeen ten, Dundee nine, Leith seven; and whalers were also sailing out of Greenock, Kirkcaldy and Montrose. [...]

Sad events lie behind this most spirited of whaling songs. By the 1820s the relatively milder northern waters were fished clean, and whalemen were having to search in more distant corners of the Arctic, notably round the mighty and bitter Melville Bay in Northwest Greenland. In 1830, a fleet of fifty British whaleships reached the grounds in early June, a month before they expected. But the same winds that had helped them, also crowded the Bay with icefloes, and locked most of the fleet in, including the Diamond, the Resolution, the Rattler (not Battler) of Leith (not Montrose), and the Eliza Swan. Twenty fine ships were crushed to splinters, and many bold whalermen froze or drowned. The Eliza Swan was among those that got free and brought the sad news home. Our song must have been made only a season or two before that tragedy, for the Diamond's maiden voyage was only in 1825. One wonders if the man who made the song was up in Melville Bay, the year of the disaster, and whether he was lost with his ship. (Notes A. L. Lloyd, 'Leviathan!')

More or less the same info can be found in sleevenotes by Michael Pollard and Eric Winter, and in a little book on Scottish whaling, The Whale Hunters by Captain Smith. Peter Hall disagrees, however:

[1995:] [Diamond Ship] was written about the first whaling voyage to the Greenland Sea in 1812, when Captain Gibbons, a member of the family that owned part of the Aberdeen whaling fleet, commanded the vessel. She was commissioned in Quebec in 1801 and ended her days crushed by the ice in 1819, when the crew stayed late in the season to augment their catch and became trapped by the advancing winter. The men were saved by crossing the ice to board a sister ship. (Peter Hall, notes 'Folk Songs of North-East Scotland')

Maybe he's talking about a different ship?