Prince of Anhalt-Dessau, son of Leopold II Maximillian of Anhalt-Dessau (1700-1751) and Giesela Agnes of Anhalt-Köthen (1721-1751). Married (1774) to Henriette of Lippe-Biesterfled-Weissenfeld (1753-1795). He had no legitimate children but did have one illegitimate son with Anna Luise Franke, named Gustav Adolf von Heideck.
Boswell was presented to Prince Albrecht at the court of Dessau on September 24, 1764. He noted that Albrecht and his brother Johann Georg (Jean George, in Boswell's journal) "spoke only German, so that I had difficult conversation of it. However, I did so well that I was liked, and I was honoured with the distinction of being offered one of the Prince's horses to go a-hunting with, which I most thankfully accepted". On the next day, they went hunting, together with the rest of the Court of Anhalt-Dessau, and Boswell "kept by the young Prince Albert, who cut the wood by neat, pretty roads which are made in different places."