Rapp, 46, told Buzzfeed News that he attended the party on his own after befriending Spacey and was the only non-adult there. After getting bored, Rapp went into a bedroom and watched TV until wee hours, when he realised no one else was left in the apartment.
Rapp said Spacey soon entered the bedroom, picked him up, and climbed on top of him in a sexual advance.
The actor said Spacey "picked me up like a groom picks up the bride over the threshold. But I don't, like, squirm away initially, because I'm like, 'What's going on?' And then he lays down on top of me."
Rapp said that he is not in touch with Spacey since the incident, but has told the story to several friends and family members.
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Soon after the revelation, Spacey apologised to Rapp saying that he "does not remember" any such encounter but he is "sorry" if he behaved in such a manner under the influence of alcohol.
"I have a lot of respect and admiration for Anthony Rapp as an actor. I'm beyond horrified to hear his story.
"I honestly do not remember the encounter, it would have been over 30 years ago. But if I did behave as he describes, I owe him the sincerest apology for what would have been deeply inappropriate drunken behaviour, and I am sorry for the feelings he describes having carried with him all these years," he posted on Twitter.
"I know that there are stories out there about me and that some have been fueled by the fact that I have been so protective of my privacy.
"As those close to me know, in my life, I have had relationships with both men and women. I have loved and had romantic relationships with men throughout my life, and I choose now to live as a gay man. I want to deal with this honestly and openly and that starts with examining my own behaviour," he added.