His work Theorie der algebraischen Functionen einer Veränderlichen (with Dedekind) established an algebraic foundation for Riemann surfaces, allowing a purely algebraic formulation of the Riemann–Roch theorem.

Weber's research papers were numerous, most of them appearing in Crelle's Journal or Mathematische Annalen.

Weber also taught in Zürich at the Federal Polytechnic Institute (today the ETH Zurich), at the University of Königsberg, and at the Technische Hochschule in Charlottenburg (today Technische Universität Berlin).

In 1893 in Chicago, his paper Zur Theorie der ganzzahligen algebraischen Gleichungen was read (but not by him) at the International Mathematical Congress held in connection with the World's Columbian Exposition.

His doctoral students include Heinrich Brandt, E. V. Huntington, Louis Karpinski, and Friedrich Levi.