Benveniste was born in New York and settled in England in the 1950s.

After the second world war Benveniste, at this time known as Albert, lived in Paris and in 1948 co-founded the Zero Press with George Solomos (who was then known as Thermistocles Hoetis). Their first publication in spring 1949 was Zero Magazine [1].

Following the second issue, which featured work by Paul Bowles, James Baldwin and Matta, Benveniste moved to London.

Besides being a poet, he also worked as a printer, a typographer, and as a book designer. In London during 1965, he co-founded and managed the Trigram Press, which published work by Tom Raworth, Jack Hirschman, J. H. Prynne, David Meltzer, B S Johnson, Gavin Ewart and Lee Harwood amongst others. Trigram Press Archive

In the 1980s Benveniste and his partner Agnetha Falk moved to Hebden Bridge in West Yorkshire where they operated a secondhand bookshop. When he died in 1990, Benveniste was buried in the graveyard of Heptonstall church.