He attended a Jewish school in Rome until the end of 1939 when his father was sponsored by members of the Menninger family to emigrate to America.

Weiss studied at the University of Chicago, where he received in 1951 his master's degree and in 1956 under Antoni Zygmund his PhD with thesis On certain classes of function spaces and on the interpolation of sublinear operators.

in 1960 he was a visiting professor in Buenos Aires and in the same year a postdoc at the Institut Henri Poincaré in Paris.

In the academic year 1987–88 he was at MSRI as the organizer of a program in classical analysis.

In 1967 Weiss won the Chauvenet Prize for his article Harmonic Analysis.