CNN independently obtained a copy of the full deposition.
Cosby, 78, has never been criminally charged and has vehemently denied wrongdoing. In the deposition, he says the sex and drug-taking were always consensual.
When reached for comment Sunday, Cosby publicist Andrew Wyatt said, "No comment at this time." Constand also said she had no comment.
And Constand attorney Dolores Troiani said she could not comment due to the confidentiality of the lawsuit settlement. CNN has not been able to obtain Constand's deposition.
In the 2005 deposition, taken in a Philadelphia hotel, the wealthy and popular entertainer faced intense questioning about how he conducted relationships with women. For instance, Troiani asked about a model he met more than two decades ago at a club in Denver.
Cosby: We had sex and we had dinners and sex and rendezvous.
Troiani: What are rendezvous?
Cosby: Rendezvous is when you call somebody and say, do you want to be at such and such and they say yes and you go there.
Troiani: Is there sexual contact associated with the rendezvous?
Cosby: There was with (the woman) every time.
The woman alleged Cosby drugged a cup of coffee and had sex with her while she was unconscious.
Troiani: She says that she was in a car and that when she awoke her clothes were a mess, her bra was undone, her top was untucked and I was sitting there going, oh, my God, what happened. Do you recall anything such as that ever happening with her?
Cosby: I wasn't there.
Earlier this month, a judge released a memorandum of law relating to the Constand lawsuit
which contained portions of the deposition in which Cosby admitted to getting prescription Quaaludes to give to young women he wanted to have sex with.
In those earlier documents, Cosby says he gave Constand one and a half tablets of Benadryl -- an over-the-counter antihistamine that can cause drowsiness -- to relieve stress.
Cosby says he obtained a prescription for the sedative Quaaludes from a Los Angeles doctor, ostensibly for a bad back. But Cosby said he never used the drug because it made him sleepy.
"Quaaludes happen to be the drug that kids, young people were using to party with and there were times when I wanted to have them just in case," he said.
For instance, Cosby said he was introduced to a young woman at the Las Vegas Hilton in the 1970s.
"She meets me backstage," he said. "I give her Quaaludes. We then have sex. ... I think she may very well have been very happy to be around the show business surroundings."
Constand's lawyer said, "She said that she believes she was not in the position to consent to intercourse after you gave her the drug. Do you believe that is correct?"
"I don't know," Cosby replied.
"Why don't you know?
"That's her statement," he said. "I don't know. How many years ago are we talking about? Nineteen seventy what?"
Twenty years later the woman contacted Cosby and asked for money, saying he'd promised her $500 for every A she earned in school, Constand's lawyer says in the deposition.
Cosby said he thought the woman was broke and gave her a total of $10,000.
When Constand's lawyer asked who Cosby wanted to hide the payments from, he replied: "Mrs. Cosby."
Cosby said he imagined his wife would have known he was also giving Constand money, to pay for her education, but not because they'd had sex -- and Constand was now upset, he said in the deposition. In her lawsuit, Constand alleged she was the victim of nonconsensual sex.
In the deposition, Cosby mostly discusses his relationship with Constand but describes sexual relationships with at least five women in different cities across the nation, in hotels and in one of his homes.
Constand was a staffer for the women's basketball team at Cosby's alma mater, Temple University, when she visited Cosby's Pennsylvania home in 2004. She told police in her home province of Ontario in January 2005 that Cosby gave her medication that made her dizzy and she woke up to find her bra undone and her clothes in disarray.
In the deposition, Cosby says he first met Constand at a Temple athletic facility where he showed her a back-relaxation technique.
"It's one where she gets on my back with her back and her arms come under mine and I grab her and I say, now you relax," the deposition quotes him as saying. "And her legs are hanging out in the air and I go up and down and I give that jolt and it's supposed to line the vertebrae."
She talked about wanting a tighter butt, he said, so he touched it.
"It wasn't a sexual opening," he said. "It's a woman who is working with basketball players, who used to be a jock, who also got on my back and is like that and she was very comfortable about that."
When asked how he caused a romantic relationship to develop, Cosby said he acted as a mentor, "Inviting her to my house, talking to her about personal situations dealing with her life, growth, education access and thoughts to how to acquire a more aggressive attitude, protecting oneself in business."
Cosby said he always looked for nonverbal clues that women welcomed his sexual advances.
"I think I'm a pretty decent reader of people and their emotions in these romantic sexual things, whatever you want to call them," he said in the deposition.
Though sometimes argumentative and seemingly evasive with Constand's lawyer, Cosby described a sexual encounter in such detail that it took several deposition pages. He said he and Constand were sitting on a sofa when he touched her stomach.
"I don't hear her say anything," he said. "And I don't feel her say anything. And so I continue and I go into the area that is somewhere between permission and rejection. I am not stopped."
Constand went to authorities with her allegations in January 2005, but in February of that year authorities -- citing a lack of evidence -- said no charges would be filed against Cosby.
Cosby eventually settled Constand's suit in which 13 "Jane Does" had similar stories of sexual abuse. The suit was settled under confidential terms in 2006.
Since then, more than 25 women have publicly accused
Cosby of raping or assaulting them over the past 40 years. He has never faced criminal charges, partly because almost all the accusations fall outside statutes of limitations.
Cosby talked about his philosophy of sex in the deposition, saying he tried to avoid sexual intercourse because he didn't want women falling in love with him.
Cosby accuser: His claim 'makes me sick'
Barbara Bowman, who said Cosby raped in her in the 1980s, said she was relieved to see the deposition come to light.
"The first thing that came to my mind was 'finally,'" Bowman told CNN's "New Day" on Monday. "After all of the years that I have been screaming my story ... it was a wonderful day."
But she said she was disgusted by Cosby's claim in the documents that sex was consensual.
"It makes me sick, and it sends a very wrong message about what consent really means," Bowman said.
"Consent is not the absence of a 'no.' And when you put drugs and alcohol in the mix, and put a person into a state of impairment and incapacitation where they're not able to respond in a normal setting, it's manipulative. It's diabolical. It's controlling. And it's typical behavior and thinking of a sociopath."