Weber was born in London, England, to Susan Ann Caroline (née Coriat), a British socialite, and husband Thomas Evelyn "Tommy" Weber (originally Thomas Ejnar Arkner), a racing driver who also came from a wealthy family. His father was born in Denmark, of Danish and English descent. Weber's maternal grandfather, Robert Coriat, who was born in Mogador, Morocco, was of Sephardic Jewish (Moroccan-Jewish) descent. Weber's maternal grandmother, Priscilla Weigall, was English, from an upper-class family. Priscilla's father was Sir William Ernest George Archibald Weigall, 1st Baronet, while Priscilla's maternal grandfather was Sir John Blundell Maple, 1st Baronet. Weber has one sibling, a brother Charley. Through his English maternal grandmother, Weber is a great-grandson of politician Sir William Ernest George Archibald Weigall, 1st Baronet and his wife, Grace Emily ( Blundell Maple), and a great-great-grandson of business magnate Sir John Blundell Maple, 1st Baronet and wife, Emily Harriet Merryweather.
Weber's mother, Susan, was diagnosed with depression and LSD-induced schizophrenia, and died at age 27 of a drug overdose when Weber was eight years old. His father, who sold various illegal drugs and utilized both his sons in trafficking, sold drugs to numerous international destinations and struggled with substance addiction until his death in 2006 at age 67.
In 1971, Weber's father took him and his brother to stay for a period at Villa Nellcôte, where the Rolling Stones were recording Exile on Main St. In a 2010 article for The Times, Weber recalled that his "father used him as a drug mule to bring cocaine out for Mick and Bianca Jagger's wedding."
Weber attended Summerhill School, Leiston, Suffolk. Later, he went to the United States to study at Middlebury College in Vermont, where he sang a cappella with the Dissipated Eight and majored in English literature and political science, graduating with a B.A. cum laude in 1986. He attended The Juilliard School's Drama Division as a member of Group 19 (1986–1990), which also included Laura Linney and Jeanne Tripplehorn. He also studied at Russia's famed Moscow Art Theatre.
At the 2010 Cannes film festival, as part of the Directors' Fortnight at the launching of the rock 'n roll documentary, Stones in Exile, singer Mick Jagger spoke to the crowd about the months of drug-fuelled recording sessions that produced the Stones' classic 1972 album Exile on Main Street. Jagger joked about the rarely seen original footage that reveals eight-year-old Weber rolling marijuana joints for them. Weber has reportedly stated that his drug-dealing father brought him to Keith Richards's rented French villa, Nellcôte, in the seaside town of Villefranche-sur-Mer near Nice, where the Stones were recording the album.
