The board's vote, which was expected, clears the way for a strike vote by the union's membership on Saturday, between the matinee and evening performances.
The vote cleared the bill for President Clinton, who has said he will veto it.
The vote would clear the way for the 10 representatives of the group to be seated.
The vote cleared the way for the rules to take effect May 17.
A vote on the bill April 27 cleared the way for debate but did not approve it.
The vote cleared the way for approval of the nomination by the full Senate, possibly next week.
The vote should clear the way for a $434 milion payment to New York City next week.
The vote on the bill cleared the way for final action this year once House and Senate negotiators work out differences in the two measures.
That vote would clear the way for South Africans to compete anywhere in the world, including the Olympics for the first time since 1960.
The legislative vote cleared the way for final approval on Wednesday by a state oversight board.
