Berg was born in New York City and educated at Pennsylvania State University and Western Reserve, where he obtained his PhD in 1952. He taught first at the School of Medicine at Washington University, St. Louis, moving to the University of Stanford in California in 1959, where he was professor of biochemistry from 1959 to 1970, and Willson professor of biochemistry from 1970.

In 1955 Francis Crick proposed his adaptor hypothesis, in which he argued that amino acids did not interact directly with the RNA template but were brought together by an adaptor molecule. Crick offered little information on the nature of such molecules, merely arguing that they were unlikely to be large protein molecules and suggesting that there might well be a specific adaptor for each of the 20 amino acids. In 1956 Berg successfully identified such an adaptor, later known as transfer RNA, even though he was then unaware of Crick's hypothesis. He found a small RNA molecule that appeared to be quite specific to the amino acid methionine.

Berg's name later became known to a much wider public with the publication in Science (24 July, 1974) of the ‘Berg letter’, written with the backing of many leading molecular biologists, in which he gave clear warning of the dangers inherent in the uncontrolled practice of recombinant DNA experiments. It had become possible, Berg stated, to excise portions of DNA from one organism, using specialized enzymes, and to insert them into the DNA of another organism. For example, the harmless microorganism Escherichia coli, found in all laboratories, could be implanted with active DNA from the tumor-causing virus SV 40 and perhaps allowed to spread throughout a human population with quite unpredictable results. Berg consequently proposed an absolute voluntary moratorium on certain types of experiment and strict control on a large number of others. An international conference was held in Asilomar, California, followed by the publication of strict guidelines by the National Institutes of Health in 1976. That such agreement could be reached and maintained, it has been claimed, was largely a result of the integrity and authority of Berg. Ironically Berg was awarded the Nobel Prize for chemistry in 1980 for the large part he played in developing the splicing techniques that made recombinant DNA techniques possible in the first place.