The jury announced in its verdict that Rapp - who demanded 40 million dollars (roughly SEK 400 million) in damages - had failed to prove that Spacey "touched a sexual or intimate part of the body".
The outcome led to the judge deciding to drop the case.
Rapp had sued Spacey in a civil suit.
It was five years ago that the first accusation against Kevin Spacey of sexual abuse came.
Anthony Rapp in an interview accused Spacey of a sexual assault in the mid-1980s.
According to Rapp, the then 26-year-old Spacey picked him up, placed him on the bed and himself on top, with his clothes on.
Spacey denied the allegations when they came.
More lawsuits are ongoing
Before the trial, Antony Rapp, who is also an actor, had requested damages and compensation for financial loss because he says he has been unable to work due to post-traumatic stress after the incident.
More lawsuits are pending against Spacey, whose successful acting career was destroyed when allegations of abuse were leveled against him during the metoo fall of 2017. As recently as this summer, Kevin Spacey had to defend himself in an English court against allegations of multiple cases of sexual assault against three men between the years 2005 and 2013. The Hollywood star has always denied the charges and that trial resumes in London next year.
Kevin Spacey was successful with the series "House of cards" on Netflix, and has also received two Oscar statuettes, for his roles in "The Suspects", in 1995, and "American beauty" four years later.