Branch attended Alabama State University where he played college baseball and basketball for the Hornets.

In 1962, after Branch enjoyed a second consecutive successful season with the Double-A San Antonio Missions — recording 216 strikeouts in 237 innings pitched[3] — the Cubs traded him to the Cardinals on September 1 for right-handed pitcher Paul Toth.

Seventeen days later, Branch made his MLB appearance as the Cardinals' starting pitcher — against Toth and the Cubs at Wrigley Field.

[4] He left the game for a pinch hitter, Red Schoendienst, in the top of the sixth inning with St. Louis trailing, 3–1.

Branch made the Cardinals' 40-man spring training roster in 1963[5] but was sent to the Triple-A Atlanta Crackers for the full season.