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Minnesota's state Supreme Court will hear arguments Thursday in a case brought by the state Democratic Party, which is suing Secretary of State Mary Kiffmeyer over her office's handling of the substitution of deceased Sen. Paul Wellstone's name on election ballots.

Former Vice President Walter Mondale accepted Wednesday evening the party's nomination of him to succeed Wellstone, who died in a plane crash last Friday. He will run against former St. Paul Mayor Norm Coleman.

Under the state's current plan, people who have already mailed in their ballots and want to change their vote will have to go to a local election office and get a new ballot, something Democrats argue is not possible for many absentee voters.

"It's not the way we're supposed to be doing things in Minnesota. We don't want to be another Florida," said state Democratic Party attorney Alan Weinblatt.

The lawsuit seeks a court order allowing voters to have a new absentee ballot mailed to them promptly if they want to make a change.

"When the DFL [Democratic-Farmer-Labor] Party submits the name for filing, which we expect on Thursday, to fill the vacancy, a nomination, only at that time do we have the authority to put the name on that ballot," Kiffmeyer said.

Democrats would prefer that absentee votes for Wellstone be automatically transferred to Mondale, contradicting the plan created by Kiffmeyer, a Republican, and Attorney General Mike Hatch, a Democrat.

"We can't assume that just because Paul Wellstone's name is here now that [voters] automatically want to transfer to whoever the new nominee is. We can't assume that. We can't read those minds," Kiffmeyer said.

That bothers the usually talkative independent Gov. Jesse Ventura. Through his aides, Ventura said the only fair thing to do is to count the absentee votes for Mondale. If not, he fears that more lawsuits will delay certification of the election.

Ventura plans to appoint someone to fill out Wellstone's term, which expires in January. he said on Wednesday that he had been planning on appointing a Democrat, but after walking out of the Wellstone memorial on Tuesday night because of its obvious political rallying, he is leaning toward an Independent.

At least two counties are breaking ranks with the state plan for absentee ballots. Officials in Ramsey County, which includes St. Paul, and Hennepin County say they will mail a new ballot to those who ask.

"We're not going to question someone as to why their ballot is spoiled," said Darwin Lookingbill, director of the county attorney's civil division. "If they call we're going to send them another ballot."

To accommodate the Mondale substitution at the polls a special supplemental ballot will be handed out covering just the U.S. Senate race.

Weinblatt adds that the extra ballot is confusing.

"Because of the shortness of time, and because of the confusion that has developed over the weekend, I am asking that ballot instructions be put on the ballot in English, in Spanish, in Hmong and in Russian," Weinblatt said.

Mike Erlandson, chief of the state Democratic party, also complained Wednesday that hand-counting of the Senate race, decided by Kiffmeyer, is inappropriate.

"That makes absolutely no sense to us," he said. "There certainly is plenty of time, after the Democratic party has a nominee tonight to supply the counties in the state of Minnesota that have the optical reading devices with appropriate ballots so that it can be done by machine and this process can be done quickly and more accurately."

The legal battle to be played out in front of the state Supreme Court Thursday is expected to be the first round in what could be several fights likely to spill over after Election Day.

Many say they are hoping to avoid a situation similar to Florida in 2000, where the presidential election was left undecided for 36 days, but several analysts say it could become an extended battle.

Fox News' Steve Brown and The Associated Press contributed to this report.