Rhame was born in Atlanta to Lee and Rachelle Rhame, one of two children and their only son. Rhame attended John H. Guyer High School in Denton, Texas, and committed to the University of Oklahoma to play college baseball. In his freshman year with the Oklahoma Sooners baseball team, Rhame gained 60 pounds (27 kg), lost velocity on his fastball and posted a 7.20 earned run average (ERA). He was cut from the team and transferred to Grayson County College. Rhame had signed on to transfer to the Texas State Bobcats for the 2014 college baseball season.\n', '

The Los Angeles Dodgers selected Rhame in the sixth round of the 2013 Major League Baseball draft. He signed with the Dodgers and made his professional debut with the Ogden Raptors. In 2014, Rhame played for the Great Lakes Loons. He appeared in 51 games and had a 2.01 ERA, 90 strikeouts and nine saves over 67 innings. Rhame was assigned to the Rancho Cucamonga Quakes of the California League to start the 2015 season. He was later promoted to the AA Tulsa Drillers, where he was 3–3 with a 3.06 ERA in 39 games. The Dodgers invited him to major league spring training in 2016 and assigned him to the Triple-A Oklahoma City Dodgers to start the season. He was 1–7 with a 3.29 ERA in 54 appearances and recorded seven saves. The Dodgers added him to their 40-man roster after the season.. He appeared in 54 games for Oklahoma City and was 1–7 with a 3.29 ERA.\n', '

On August 20, 2017, the Dodgers sent Rhame to the New York Mets as the player to be named later in the trade that sent Curtis Granderson to the Dodgers. The Mets promoted Rhame to the major leagues on September 1. Rhame made his Major League debut on September 2, 2017 in a scoreless outing against the Houston Astros at Minute Maid Park.\n', '

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