Usually the third holds the broom while the skip throws. That's why the third is also called the "vice skip." The third holds the broom because he or she is usually the next most experienced person on the team, after the skip.
Really, though, there aren't any rules about who must hold the broom, when. The lead could hold the broom the whole game (except when he's shooting, of course) if that's what the team wanted.
On some teams, the third might actually be better at "calling ice" (showing where to aim) or making sweeping calls, so they might have the third hold the broom throughout the game, instead of the skip.
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