Adcock was born in Coushatta, Louisiana. He was drafted by the Cincinnati Reds after a successful run at Louisiana State University, but Ted Kluszewski had firm hold of the team's first base slot. Adcock played in left field from 1950-52, but was unhappy and demanded a trade, which he received. His first season with the Braves was capped by a mammoth home run into the center field bleachers at the Polo Grounds on April 29, 1953, a feat that had never been done before and would only be accomplished twice more, by Hank Aaron and Lou Brock. On July 31, 1954, he accomplished the rare feat of homering four times in a single game, against the Brooklyn Dodgers at Ebbets Field, and set a new record for most total bases in a game (18) that stood until broken by Shawn Green in 2002. Another notable home run was the blast that ended the epic duel between Lew Burdette and Harvey Haddix on May 26, 1959, in which Haddix took a perfect game into the 12th inning. Adcock did not get credit for a home run, however, because Aaron, who was on first base, saw the runner ahead of him score the winning run and thought the hit had only been a double and walked back to the dugout, causing Adcock to be called out for passing him on the basepaths.

Adcock was often overshadowed both by his own teammates Aaron and Eddie Mathews, and by the other slugging first basemen in the league, Kluszewski and Gil Hodges, but he did make one All-Star team (1960) and was regularly amongst the league leaders in home runs. In 1956, he finished second in the National League in home runs, runs batted in, and slugging average. After playing for the Cleveland Indians (1963) and Los Angeles/California Angels (1964-66), Adcock managed the Cleveland Indians for one year (1967) and managed two more years in the minor leagues before settling down at a 288-acre (1.2 km²) ranch in Coushatta, Louisiana to raise horses. He died in Coushatta at age 71.