The Star Tribune polls Minnesotans one year before they go to vote on the marriage amendment:

Would you favor or oppose amending the Minnesota constitution to allow marriage only between a man and a woman?
Favor 48%
Oppose 43%
Don't know/Refused 8%

This contradicts earlier polls which suggested the amendment enjoyed less support, as the Tribune also reports:

The poll may reflect some shifting opinions among Minnesota voters. In May of this year, just before the constitutional question was approved by the Legislature, 55 percent of respondents in a Minnesota Poll said they opposed the amendment while 39 percent favored a constitutional ban on gay marriage. That survey, like the one conducted last week, also reflected sharp demographic and partisan splits.

Amendment supporter Chuck Darrell, communications director for Minnesota for Marriage, said the early signs of support reflected in the recent Minnesota Poll was good news for his side. He said even though the lead in favor of the amendment is slim, it may reflect even more support.

Darrell said in other states, notably California and Maine, early polling showed marriage amendments failing. But when voters are in the secrecy of their voting booths, he said, they supported the amendments.

"People tend to give the politically correctly answer" on marriage amendment polls, he said.