Carne was born in Northampton, England. Her parents, Harold and Kathy, were greengrocers in Kingsthorpe. She made her first British television appearances on the series Danger Man (1961) and episodes of The Rag Trade (also 1961), a BBC sitcom. She moved to the USA not long afterwards. Her first regular role was in the sitcom Fair Exchange (1963) as an English teenager who goes to the US to live with an American family, whose daughter (played by Lynn Loring) has gone to live in England. That was followed by The Baileys of Balboa (1964). She later co-starred with Pete Duel in Love on a Rooftop (1966). She made several appearances on the adventure series The Man from U.N.C.L.E..
Carne was married to actor Burt Reynolds from 1963 to 1965 and to Producer Robert Bergmann from 1970 to 1971. Both marriages were brief, childless, and ended in divorce. In 1978, after being found not guilty of possessing heroin, she was involved in a car accident along with her second husband; she recovered from a broken neck. Her drug Problem continued and she was later arrested again for heroin possession.
She had a small part in the ninth episode of the TV series Gidget (1965), guest-starred as Jill in first-season episode 2, "Follow the Leader" and as Floy in second-season episode 3, "Then Came The Mighty Hunter" of 12 O'Clock High (1965), and appeared in an episode of I Dream of Jeannie (1966). She appeared in the Bonanza episode "A Question of Strength" (1963) as Sister Mary Kathleen, two episodes of The Big Valley (1967), guest-starred in episode 11 of the first season of Alias Smith and Jones (1971) and the TV adaptation of QB VII (1974). Her film roles included A Pair of Briefs (1962), The Americanization of Emily (1964), the wife of Tom Bell in All the Right Noises (1971), and Rachel Amodeo's street movie What About Me (1993), opposite Richard Hell and Johnny Thunders.
On Rowan & Martin's Laugh-In (1968–1970) Carne gained stardom. Her most popular routine ended with her saying "Sock it to me!", at which point she was doused with water or assaulted in some other way. Carne was a regular in the first two seasons (1968–69); then, having decided the show had become "a big, bloody bore", made occasional guest appearances in the remaining 1969–70 seasons.
Carne starred in a revival of the musical The Boy Friend which opened on Broadway on 14 April 1970 and ran for 111 performances.
Her autobiography, Laughing on the Outside, Crying on the Inside: The Bittersweet Saga of the Sock-It-To-Me Girl (1985), chronicled her difficulties with drugs, her failed marriage to Reynolds, and her bisexuality.
Carne moved back to Northamptonshire, England, in the 1980s, living quietly in the village of Pitsford. She died from pneumonia on 3 September 2015 at a hospital in Northampton.