In 2001 and 2002, Weber was a series regular in HBO's The Mind of the Married Man and made guest appearances on Law & Order: Criminal Intent and NYPD Blue.

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 (née Blundell Maple), and a great-great-grandson of business magnate Sir John Blundell Maple, 1st Baronet and wife, Emily Harriet Merryweather.

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.

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.

[11] Weber's roles were often bit parts in A-list films, beginning with that of Kyra Sedgwick's character's unnamed boyfriend in the Oliver Stone-directed period saga Born on the Fourth of July (1989) and continuing with work for such directors as Sidney Lumet (A Stranger Among Us, 1992), Alan J. Pakula (The Pelican Brief, 1993) and Martin Brest (Meet Joe Black, 1998).

He scored one of his premier leads as Dr. Matt Crower, a kindly physician who takes charge of a young boy and protects him from a possessed sheriff in actor-turned-producer Shaun Cassidy's short-lived, but well received, supernatural drama series American Gothic (1995) on CBS.

That programme did not last long; and neither did the Mike Binder sitcom The Mind of the Married Man (2001), in which Weber played one of the leads, Chicago newspaper employee Jake Berman.