To simplify the thread - I am posting the actual rule from the book

ART. 4... A hitter may be (not mandatory) designated for any one starting player (not just pitchers) and all subsequent substitutes for that player in the game. A starting defensive player cannot be listed as the designated hitter in the starting lineup. A designated hitter for said player shall be selected prior to the start of the game, and his name shall be included on the lineup cards presented to the umpire-in-chief and to the official scorer. Failure to declare a designated hitter prior to the game precludes the use of a designated hitter in that game. If a pinch hitter or pinch runner for the designated hitter is used, that player becomes the new designated hitter. The player who was the designated hitter may re-enter as the designated hitter under the re-entry rule. A designated hitter and the player for whom he is batting are locked into the batting order. No multiple substitutions may be made that will alter the batting rotation. The role of the designated hitter is terminated for the remainder of the game when:

a. the defensive player, or any previous defensive player for whom the designated hitter batted, subsequently bats, pinch-hits or pinch-runs for the designated hitter; orb. the designated hitter or any previous designated hitter assumes a defensive position.

I believe the answer to cshs81 is in bold red.