Number the balls from 1 to 9. Weigh {1,2,3} against {4,5,6}. If one set is heavier, the heavier ball is among those three balls. If they're of equal weight, then the heavier ball is among {7,8,9}.

To find the heavier ball among, say, {1,2,3} only weighing once, we weigh {1} against {2}. If either one is heavier than the other, that one is the heavier ball. If they're of equal weight, {3} is the heavier ball.

For these problems, it helps to reason backwards. Start at the end: from how many balls can I discern the heavier one if I can weigh only once? Then extend to weighing twice, and so on.