The poll, by Public Policy Polling (PPP), said 48 percent of Minnesotans support the amendment and 47 percent oppose it, with less than sixty days to go until the November election.
State law already bans same-sex marriage.
"In January we found 48/44 support for the ban, while in June we found 49/43 opposition," said a statement from the pollsters, published on PPP's website. "It looks like a toss up."
Opinions broke down by age group thus:
Women (52/41), Democrats (78/16), and voters under 45 (50/45) all oppose the ban. Men (55/41), Republicans (80/17), independents (51/42), and seniors (53/40) all support it in greater numbers though.
For their poll, PPP surveyed 824 likely Minnesota voters, on Sept. 10 and 11. The poll's margin of error was given as plus or minus 3.4 percent.
A , said that 50 percent of Minnesotans support the amendment and 43 percent oppose it, with eight percent remaining undecided.