Weisberg was born in New York City. He attended the Fiorello H. LaGuardia High School of Music & Art and Performing Arts and later the Juilliard School, majoring in bassoon and studying with Simon Kovar.

Soon after graduation he found notable success securing the principal chairs with the Houston and Baltimore Symphony Orchestras and second bassoon with the Cleveland Orchestra, before coming back to New York City. After pursuing study of conducting with Jean Morel he again returned to the bassoon as principal for Symphony of the Air as well as bassoonist of the New York Woodwind Quintet for 14 years.

In the realm of conducting he has conducted the Milwaukee Symphony Orchestra and the Sjaellands and Aalborg Symphonies of Denmark. Weisberg founded and conducted the Contemporary Chamber Ensemble.

Weisberg also extensively taught, having held posts at the Juilliard School, State University of New York at Stony Brook, Yale School of Music, and Indiana University's Jacobs School of Music.

He recorded several renderings of 20th century music, i.a. from Schoenberg (Pierrot Lunaire), Varèse, Messiaen, and contemporary American composers (i.a. Carter, Babbitt, Crumb), mostly with the Contemporary Chamber Ensemble and the Ensemble 21. These met with considerable critical acclaim.

Weisberg also authored two books: The Art of Wind Playing (G. Schirmer) and Performing 20th Century Music-a Handbook for Conductors and Instrumentalists (Yale University Press) as well as composed and edited numerous scores, including the Bach Cello Suites.

Weisberg introduced a new bassoon which he claimed was "absolutely unable to crack. Perfect slurs on the most problematic notes. Never having to flick again. Ease of fingering. Better quality and pitch on six of the worst notes.This is what the Weisberg System promises and delivers, and it does all of this automatically, with no new fingerings to learn."