This clip is actually an 8mm home movie from 1956.
Oliver Hardy was already sick at the time, and the doctors told him to loose weight fast, which he did as you can see. Shortly after this film was shot, he had another heart attack (some say brought on by the rapid weight loss) which completely debilitated him, and he died some months later in August 1957.
The last professional appearance of Laurel and Hardy was on the BBC Television show "This Is Music Hall" in 1955, just before Hardy got sick.
They filmed a little "hello" in Los Angeles to the Water Rats (the British theatrical fraternity who were a part of the show) and the film was inserted into the TV show.
Here it is, slightly edited. It is their last public appearance together. A perfect "goodbye".