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Weinstein had been expected to move to New York City's notorious Rikers Island jail complex following the verdict, but was admitted late on Monday night to Bellevue Hospital a few kilometres away.
His lawyer Arthur Aidala told reporters after meeting with Weinstein that his 67-year-old client "looked like he was in good shape" but was "not a picture of health by any stretch of the imagination," and doctors would decide when to release him.
"He was upbeat," though "obviously he prefers being in his own house," Aidala said.
Media reports said Weinstein had experienced chest pain or heart palpitations before being admitted to Bellevue.
Aidala said Weinstein was "somewhat flabbergasted by the verdict" and "cautiously optimistic" about his eventual appeal. "I almost feel emboldened by Mr Weinstein's spirits and his desire to continue to fight these charges," Aidala added.
On Monday, Weinstein was convicted of sexually assaulting former production assistant Mimi Haleyi in 2006, and raping former aspiring actress Jessica Mann in 2013.
Jurors acquitted Weinstein on the two top charges, predatory sexual assault, which carried a maximum life sentence.
The guilty verdict was a milestone for the #MeToo movement, which Weinstein's case fueled in late 2017 and inspired women to accuse hundreds of powerful men in entertainment, business, media, politics and other fields of sexual misconduct.
Weinstein faces up to 29 years in prison, and is scheduled to be sentenced on March 11.
Aidala said he may seek an earlier sentencing date so Weinstein can appeal sooner. New York law requires a defendant to be sentenced first, he said.
He quoted Weinstein as saying after the verdict: "I'm innocent. I'm innocent. How can this happen in America?"
More than 80 women have accused Weinstein of sexual misconduct. He has denied the accusations, and said any sexual encounters were consensual.
Actress Rose McGowan, who accused Weinstein of raping her, called the conviction "a huge step forward in our collective healing."
Weinstein was a key force behind acclaimed films such as The English Patient and Shakespeare in Love, which both won Oscars for best picture.
He had been free on bail during the trial, and eaten breakfast with his lawyers at a Four Seasons hotel on Monday morning.
Weinstein lost that freedom when the trial judge, Justice James Burke of Manhattan state supreme court, ordered that he be jailed following the conviction, and court officers led him away in handcuffs.
Aidala said Weinstein was not being handcuffed at Bellevue.
Weinstein was admitted to a unit that provides medical care for inmates, after media reports said he had experienced chest pain or heart palpitations.
A spokesman said earlier on Tuesday that Weinstein suffers from diabetes and high blood pressure.
Weinstein appeared frail and used a walker during the trial.
The Rikers Island jail, whose main building went up in 1932, has long been plagued by violence and neglect, and New York City Mayor Bill de Blasio vowed in 2017 to close it within a decade.
Jail officials could place Weinstein in a private or semi-private cell to help ensure his safety.
He would likely wear a tan jumpsuit, which indicates that he had yet to be sentenced, according to Malissa Allen, a mental health counsellor who has treated Rikers inmates.
Weinstein also faces several felony charges in Los Angeles in connection with his alleged sexual assaults against two women in 2013.
Dozens of women have also filed civil lawsuits against Weinstein. His former film studio, Weinstein Co, filed for bankruptcy in March 2018 and is being liquidated.
Some of the more than 80 women who have accused him of sexual misconduct say they are looking to his trial in Los Angeles to build on what they called a seismic shift in attitudes signalled by his conviction in New York.
On Tuesday, a dozen women held an emotional news conference in Los Angeles.
"Now we know that if we dare to speak, there is a far greater chance we will be heard and our abusers will be punished," actress Rosanna Arquette said.
"It is a historic shift that I never thought I would see in my lifetime," added Arquette, one of a group of Weinstein's accusers called the Silence Breakers.
In January, he was charged in Los Angeles with raping one woman and sexually assaulting another in the city in 2013. Weinstein has denied having nonconsensual sex with anyone; no date has been set for the Los Angeles case to begin.
"Now we can turn our attention to the upcoming trial here in Los Angeles. We have an opportunity to build on this momentum," Canadian actress Larissa Gomes told the news conference.
The women praised their six peers for their bravery in testifying at the New York trial, where the accusers were grilled by Weinstein's lawyers, who sought to portray them as using the movie mogul to advance their careers.
"Harvey, you messed with the wrong women," said former actress and screenwriter Louisette Geiss. "We will see you here in Los Angeles, where hopefully your conviction will leave you in jail for life."
The women reported mixed emotions on hearing about Weinstein's conviction on Monday. For some, the verdict came more than 20 years after they say they were abused by the producer.
"Many of us didn't think this day would come," said television reporter Lauren Sivan. "We were actually really nervous and holding our breath because there was a very good chance he would walk."