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 principal chair with the Houston Symphony Orchestra and Baltimore Symphony Orchestra, as well as second bassoon with The Cleveland Orchestra. After studying conducting with Jean Morel he became principal bassoon for Symphony Of The Air as well as bassoonist of the New York Woodwind Quintet for 14 years.
Weisberg has conducted the Milwaukee Symphony Orchestra, the The New York Philharmonic Orchestra, and the Sjællands Symfoniorkester and Aalborg Symfoniorkester of Denmark. Weisberg founded the Contemporary Chamber Ensemble in 1960 and conducted the ensemble. He has also 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 made several recordings of 20th century music from Arnold Schoenberg (<i>Pierrot Lunaire, Erwartung</i>), and Edgard Varèse, Olivier Messiaen; as well as contemporary American composers such as Elliott Carter, Stefan Wolpe, and George Crumb. The majority of these recordings were made with the Contemporary Chamber Ensemble, the Orchestra Of The 20th Century, and Ensemble 21.