The American League uses a designated hitter.
Some exceptions:
If an American League team plays at a National League Park (Yankees play at Shea Stadium or the new Citi Field against the Mets) then the American League team can't use a Designated Hitter.
If a National League team plays at an American League Park (Mets play at Yankees Stadium) then the National League team is allowed to use a designated hitter.
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The DH is used in all minor leagues. The only time where pitchers still bat is at the Double-A and Triple-A level, when both teams are National League affiliates; otherwise the DH is used.
In the Pacific Coast League, pitchers only hit when both clubs are NL affiliates and both clubs agree to have their pitchers hit. The reason for this is that as players move up and get closer to reaching the majors, teams prefer to have the rules follow (as closely as possible) those of the major leagues