He was the sixth (but second surviving) son of Heinrich of Saxe-Weissenfels, Count of Barby, and his wife, Princess Elisabeth Albertine of Anhalt-Dessau.
The death of his oldest surviving brother, the Hereditary Prince Frederick Heinrich, during a trip to The Hague (21 November 1711) made him the new heir of the county of Barby.
In Forst, Niederlausitz, on 18 February 1721, Georg Albrecht married Duchess Auguste Louise of Württemberg-Oels.
Auguste's maternal grandmother was Countess Justine Sophie of Barby-Mühlingen (1636-1677), a sister of the last count; this brought Georg Albrecht closer connection to the former ruling house of Barby.
The marriage was extremely unhappy, and the couple finally divorced in 1732 after eleven years of childless union.