However, that meeting actually took place in Ujiji in Tanzania on 10 November 1871 as clearly detailed in Stanley's book, "How I Found Livingstone".
Livingstone then details meetings with several Arab residents of Ujiji including one who was supposed to be keeping his goods from his previous visit, before recording Stanley's arrival.
At the Mugere River they found the village of Chief Mukamba who welcomed them and gave them a hut in which to rest.
They stayed two nights, and Stanley records that Livingstone's servant Susi got very drunk on the Chief's hospitality.
[1] As the first Europeans to visit the area, their arrival was memorable, and it must be at some time later the event became confused in some people's minds as the first meeting between Livingstone and Stanley.