Composed, unwavering, yet sometimes tearful, Ms. McDougal told a detailed story about a relationship that was based on love — “he always told me he loved me,” she said — and was facilitated by one of Mr. Trump’s bodyguards.

She described herself as a Republican and said she voted for Mr. Trump in November, describing him as “brilliant” and “charming” and someone she loved enough to even consider marrying. She said she was now eager to tell her story because it was already known and she wanted the chance to tell it herself.

Ms. McDougal said she first met Mr. Trump when he filmed an episode of “Celebrity Apprentice” at the Playboy Mansion in 2006, where they struck up conversation and he asked for her number.

She said they had their first date that June, on Mr. Trump’s birthday, at the Beverly Hills Hotel. Mr. Trump invited her for dinner, she said, so she was surprised when they entered the grounds from a rear entrance and went directly to a private bungalow. “We’re talking about his birthday, and then as the night ended, we were intimate,” she said.

What followed, she said, were dozens of dates and sexual encounters in which, she said in answer to a specific question from Mr. Cooper, Mr. Trump did not use birth control.

Many of those encounters, she said, took place at the Beverly Hills Hotel, but she also spent time with Mr. Trump at his other properties across the country. “I went to his golf course in California,” she said. “I went to his golf course home in New Jersey. I went to his home in New York.” At one party, she met his son Eric, and posed with him in a picture, she said. At another, at the Playboy Mansion, she posed for a picture with Mr. Trump, Melania Trump, Mr. Trump’s daughter Ivanka and other people, keeping her distance from Mrs. Trump as best she could, she said.

Ms. McDougal said Mr. Trump had compared her, favorably, to his daughter. “He said I was beautiful like her and, ‘You’re a smart girl,’” she said.