ALBANY – Democrats picked up another victory in the State Senate after the opening of the final paper ballots this morning in a long-contested race in Ulster County.

Democrat Cecilia Tkacyzk will represent the sprawling Hudson Valley and Mohawk Valley 46th Senate District that Republicans created during last year’s redistricting process with the idea that GOP candidate George Amedore would easily win.

But Tkaczyk capitalized on the GOP’s failure to pay much attention to the race until it was too late in October; the Albany County Democrat had, in the end, a better absentee ballot strategy than Amedore, especially in the more Democratic-leaning southern end of the district in Ulster County. That’s where the final ballots were counted this morning.

What her victory means remains to be seen.

The GOP is still in partial control of the Senate thanks to an unusual power-sharing deal with five breakaway Democrats. For now, Tkaczyk gives Democrats another vote if close or controversial matters make it to the floor, such as on campaign finance reforms or raising the minimum wage. Her presence also would matter if the coalition deal between Republicans and the breakaway Democrats stumbles.

The 46th District was created specifically for Amedore, who was in the Assembly before his Senate run. The Senate GOP added a new seat to the Senate – making it a 63-member chamber – during last year’s redistricting process that was approved by Gov. Andrew M. Cuomo. The district runs from west of Amsterdam in Montgomery County along the Mohawk River to a town a couple hours south in Ulster County.

The Republican was up by 37 votes after the Court of Appeals declined to hear his case to block further counting of 99 remaining ballots. After the Ulster County vote today, Tkaczyk was up by 19 votes with just one more left to count in Albany County.

“No one believed our campaign had a chance in a district hand-carved by Republicans, and yet the power of good ideas and a strong campaign proved itself,’’ Tkaczyk said in a statement.

Amedore conceded the race is over. “The time for politics has ended and the time to govern is at hand,’’ he said.

Bronx Sen. Jeff Klein, the leader of the breakaway Democratic group that formed an alliance with GOP senators to run the Senate, welcomed Tkacyzk, who is not a member of Klein’s organization, to the chamber after her victory.

Tkacyzk got sizable financial help from Friends of Democracy, which was co-founded by Jonathan Soros, son of billionaire and Democratic Party donor George Soros. The younger Soros backed her effort in large part because of her support for creation of a taxpayer-funded campaign finance system in New York state.

“Her win today is an unmistakable mandate to work to change the broken campaign finance laws that have shut out the voices of regular New Yorkers,’’ Soros said in a statement.

The race had been undecided since November with both sides taking the fight to three levels of state courts over what absentee and affidavit ballots could be counted.