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). Weber 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 signed on as one of the leads, Chicago newspaper employee Jake Berman.