New York stage
Meacham debuted on Broadway as Ensign Jane Hilton in 1952's The Long Watch (written by Harvey Haislip), for which she received a Clarence Derwent Award, given to outstanding newcomers to the New York stage.
She is most famous for her roles on and off-Broadway, most notably in adaptations of plays written by Tennessee Williams, who was a close friend. Williams once wrote an editorial in The New York Times praising Meacham, noting "There's nothing she won't say or do onstage without any sign of embarrassment" ().
She also portrayed roles in the Broadway productions of Candide and A Passage to India. She won two Best Actress Obie Awards, one for her role as "Catherine Holly" (the first actress to play the role which was later essayed by Elizabeth Taylor in the film version) in Tennessee Williams' Suddenly, Last Summer (1958) and another for the title role in Henrik Ibsen's Hedda Gabler (1961).
She also appeared in The Gnädiges Fraulein]] in 1966, and In the Bar of a Tokyo Hotel in 1969. Her final Broadway credit was as Queen Gertrude in Rosencrantz and Guildenstern Are Dead in 1968. All of her stage appearances after 1968 (including In the Bar of a Tokyo Hotel) were off-Broadway.
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On television, she was most famous for playing the eccentric maid, "Louise Goddard", on Another World. She played the role from 1972 to 1982. Meacham's character was most recognized for naming all of the Cory family's houseplants, which numbered well into the dozens.
The trademark of the Another World casting department was to hire heavily from the New York City stage, and it was noted by author Annie Gilbert in the book, All My Afternoons, that Meacham was one of the many cast members taken from this genre to infuse strong acting performances into the show, due to her experience.