Born in Ann Arbor, Michigan, Weller was educated at the University of Michigan, where his father was professor of pathology, and at Harvard, where he gained his MD in 1940. After serving in the US Army Medical Corps from 1942 until 1945 Weller worked with John Enders at the Children's Medical Center, Boston. In 1954 he returned to Harvard as professor of tropical public health, becoming professor emeritus in 1985.

In 1948 Weller, in collaboration with Franklin Neva, succeeded in growing the German measles (rubella) virus in tissue culture. They later went on to grow and isolate the chickenpox virus in a culture of human embryonic muscle and skin. With Enders and Frederick Robbins, Weller successfully applied the same method to the culture of poliomyelitis virus. By making adequate supplies of polio virus available to laboratory workers, this opened the way for the development of a successful polio vaccine.

For this work Weller shared the 1954 Nobel Prize for physiology or medicine with Enders and Robbins.