The Pirate Party now has the support of 7.9 percent of Swedish voters, up from 3.4 percent from a week earlier, according to the Demoskop poll conducted between May 13 and Wednesday.
That is still well behind Sweden's opposition Social Democrats with 35.9 percent and Prime Minister Fredrik Reinfeldt's conservative Moderate Party with 24.1 percent.
But it could be enough for it to clinch at least one seat in the European Parliament, on the heels of the April 17 conviction in Sweden of the three founders of the unrelated file-sharing website The Pirate Bay">Pirate Bay.
"Of course it is fantastic, but nothing is clear until the votes are counted," Pirate Party leader Rick Falkvinge told Sweden's domestic TT news agency.
"It is still up to all those who care about basic civil rights and these fundamental values to go out and vote."
Founded in 2006, the Pirate Party "wants to fundamentally reform copyright law, get rid of the patent system and ensure that citizens' rights to privacy are respected," it says on its website.
It took 0.6 percent of the vote in Sweden's last general election in 2006.
Demoskop spoke to more than 1,000 adults in Sweden for its poll, with about one in three respondents saying they were certain to cast their ballots on June 7 for the only directly-elected EU institution.