Weber was born in London, England and attended Middlebury College in Vermont, where he majored in English Literature and Political Science, graduating cum laude with a BA. He attended the Juilliard School, from which he graduated in 1991, and also studied at Russia's famed Moscow Art Theatre. He sang in Middlebury College's Dissipated Eight.
Weber's break-out role in the US came in 1993, as a lawyer informant in the thriller The Pelican Brief with Julia Roberts and Denzel Washington.