HMS Swallow began life as a Falcon-class sloop (12 guns, 2 masts) at Rotherhithe in 1745. She served as an "impress ship" for the impressment service, and like so many ships of that type she may well have been only remembered as a name on a muster had she not been selected as the escort to the HMS Dolphin on her exploration of the Pacific. She was heavily modified (she received a mizzenmast in the modifications) and in 1766 the two ships sailed off for the Pacific.
The two ships did not remain together for too long. Just after leaving the Straits of Magellan, they separated. After sailing for a time without sighting anything, Carteret promised a bottle of brandy for the first man to sight land. On July 2nd, 1767, Midshipman Robert Pitcairn sighted an island and Carteret wrote "It is so high that we saw it at a distance of more than fifteen leagues, and it having been discovered by a young gentleman, son to Major Pitcairn of the marines, we called it 'Pitcairn's Island.'" They did not land, and made a quick map, as well as noted the location (as time would prove, both proved woefully inaccurate). It is presumed that the 15 year old Robert Pitcairn enjoyed his bottle of brandy.
The Swallow, unlike the Dolphin, did not come across Tahiti, but carried on making other discoveries. Carteret rediscovered the Solomon Islands, as well as discovering and naming the Carteret Islands and the Duke of York Islands.
On March 20th,, 1769, the Swallow arrived back in Spithead, having circumnavigated the globe. She was apparently broken up that same year.
I got my information from various sources, including "Hawkesworths Voyages," "Captain Carteret and the Voyage of the Swallow" by H.G. Mowat, and the ships log.