A same-sex ceremony honors traditional African culture.
The bill, which has already been passed by the state Senate, is within about a dozen votes of the 60 needed to pass the House.
With a change of heart by one South Dakota senator, there are now three Democrats in the Senate who oppose marriage equality.
The close Clinton adviser said he had a change of heart on marriage equality since his last run in the state.
Opponents of marriage equality had sought to muddy the language that voters will see in 2014.
The centrist Maine Republican has a long record of support for LGBT rights in the Senate, and sometimes it sounds like she supports marriage equality.